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Theorem List for Metamath Proof Explorer - 27601-27700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuun2221 27601 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ph  /\  ( ps  /\  ph ) )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theoremuun2221p1 27602 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ( ps 
 /\  ph )  /\  ph )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theoremuun2221p2 27603 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ps  /\  ph )  /\  ph  /\  ph )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theorem3impdirp1 27604 A deduction unionizing a non-unionized collection of virtual hypotheses. 3impdir 1243 is ~? uun3132 and is in set.mm. 3impdirp1 27604 is ~? uun3132p1. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ch  /\  ps )  /\  ( ph  /\ 
 ps ) )  ->  th )   =>    |-  ( ( ph  /\  ch  /\ 
 ps )  ->  th )
 
16.22.4  Theorems proved using virtual deduction
 
TheoremtrsspwALT 27605 Virtual deduction proof of the left-to-right implication of dftr4 4058. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4058 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  A  C_  ~P A )
 
TheoremtrsspwALT2 27606 Virtual deduction proof of trsspwALT 27605. This proof is the same as the proof of trsspwALT 27605 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  A  C_  ~P A )
 
TheoremtrsspwALT3 27607 Short predicate calculus proof of the left-to-right implication of dftr4 4058. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 27606, which is the virtual deduction proof trsspwALT 27605 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  A  C_  ~P A )
 
Theoremsspwtr 27608 Virtual deduction proof of the right-to-left implication of dftr4 4058. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 27608 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  ~P A  ->  Tr  A )
 
TheoremsspwtrALT 27609 Virtual deduction proof of sspwtr 27608. This proof is the same as the proof of sspwtr 27608 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  ~P A  ->  Tr  A )
 
TheoremsspwtrALT2 27610 Short predicate calculus proof of the right-to-left implication of dftr4 4058. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 27609, which is the virtual deduction proof sspwtr 27608 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  ~P A  ->  Tr  A )
 
TheorempwtrVD 27611 Virtual deduction proof of pwtrOLD 27612. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  ~P A )
 
TheorempwtrOLD 27612 The power class of a transitive class is transitive. The proof of this theorem was automatically generated from pwtrVD 27611 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Moved into main set.mm as pwtr 4164 and may be deleted by mathbox owner, AS. --NM 15-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  ~P A )
 
TheorempwtrrVD 27613 Virtual deduction proof of pwtrrOLD 27614. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( Tr  ~P A  ->  Tr  A )
 
TheorempwtrrOLD 27614 A set is transitive if its power set is. The proof of this theorem was automatically generated from pwtrrVD 27613 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Moved into main set.mm as pwtr 4164 and may be deleted by mathbox owner, AS. --NM 15-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( Tr  ~P A  ->  Tr  A )
 
TheoremsnssiALTVD 27615 Virtual deduction proof of snssiALT 27616. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  C_  B )
 
TheoremsnssiALT 27616 If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 3700. This theorem was automatically generated from snssiALTVD 27615 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  C_  B )
 
TheoremsnsslVD 27617 Virtual deduction proof of snssl 27618. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( { A }  C_  B  ->  A  e.  B )
 
Theoremsnssl 27618 If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss 3689. The proof of this theorem was automatically generated from snsslVD 27617 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( { A }  C_  B  ->  A  e.  B )
 
TheoremsnelpwrVD 27619 Virtual deduction proof of snelpwi 4158. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  e.  ~P B )
 
TheoremsnelpwrOLD 27620 If a class is contained in another class, then its singleton is contained in the power class of that other class. This theorem is the left-to-right implication of the biconditional snelpw 4159. Unlike snelpw 4159, 
A may be a proper class. The proof of this theorem was automatically generated from snelpwrVD 27619 using a tools command file, translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Moved to snelpwi 4158 in main set.mm and may be deleted by mathbox owner, AS. --NM 10-Sep-2013.) (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  { A }  e.  ~P B )
 
TheoremunipwrVD 27621 Virtual deduction proof of unipwr 27622. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  C_ 
 U. ~P A
 
Theoremunipwr 27622 A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4162. The proof of this theorem was automatically generated from unipwrVD 27621 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  C_ 
 U. ~P A
 
TheoremsstrALT2VD 27623 Virtual deduction proof of sstrALT2 27624. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
 
TheoremsstrALT2 27624 Virtual deduction proof of sstr 3129, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 27623 using the command file translatewithout_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
 
TheoremsuctrALT2VD 27625 Virtual deduction proof of suctrALT2 27626. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  suc  A )
 
TheoremsuctrALT2 27626 Virtual deduction proof of suctr 4412. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 27625 using the tools command file translatewithout_overwritingminimize_excludingduplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  suc  A )
 
Theoremelex2VD 27627* Virtual deduction proof of elex2 2752. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  E. x  x  e.  B )
 
Theoremelex22VD 27628* Virtual deduction proof of elex22 2751. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  A  e.  C ) 
 ->  E. x ( x  e.  B  /\  x  e.  C ) )
 
Theoremeqsbc3rVD 27629* Virtual deduction proof of eqsbc3r 2992. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. C  =  x  <->  C  =  A ) )
 
Theoremzfregs2VD 27630* Virtual deduction proof of zfregs2 7348. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  =/=  (/)  ->  -.  A. x  e.  A  E. y ( y  e.  A  /\  y  e.  x )
 )
 
Theoremtpid3gVD 27631 Virtual deduction proof of tpid3g 3682. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A }
 )
 
Theoremen3lplem1VD 27632* Virtual deduction proof of en3lplem1 7349. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  E. y ( y  e.  { A ,  B ,  C }  /\  y  e.  x ) ) )
 
Theoremen3lplem2VD 27633* Virtual deduction proof of en3lplem2 7350. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  E. y
 ( y  e.  { A ,  B ,  C }  /\  y  e.  x ) ) )
 
Theoremen3lpVD 27634 Virtual deduction proof of en3lp 7351. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )
 
16.22.5  Theorems proved using virtual deduction with mmj2 assistance
 
Theoremsimplbi2VD 27635 Virtual deduction proof of simplbi2 611. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ( ph  <->  ( ps  /\  ch ) )
3:1,?: e0_ 27560  |-  ( ( ps  /\  ch )  ->  ph )
qed:3,?: e0_ 27560  |-  ( ps  ->  ( ch  ->  ph ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ch  ->  ph ) )
 
Theorem3ornot23VD 27636 Virtual deduction proof of 3ornot23 27286. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::
 |-  (. ( -.  ph  /\  -.  ps )  ->.  ( -.  ph  /\  -.  ps ) ).
2::  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ( ch  \/  ph  \/  ps ) ).
3:1,?: e1_ 27412  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ph ).
4:1,?: e1_ 27412  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ps ).
5:3,4,?: e11 27473  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ( ph  \/  ps ) ).
6:2,?: e2 27416  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ( ch  \/  ( ph  \/  ps ) ) ).
7:5,6,?: e12 27512  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ch ).
8:7:  |-  (. ( -.  ph  /\  -.  ps )  ->.  ( ( ch  \/  ph  \/  ps )  ->  ch ) ).
qed:8:  |-  ( ( -.  ph  /\  -.  ps )  ->  ( ( ch  \/  ph  \/  ps )  ->  ch ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  ph  /\  -.  ps )  ->  ( ( ch  \/  ph  \/  ps )  ->  ch ) )
 
Theoremorbi1rVD 27637 Virtual deduction proof of orbi1r 27287. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2::  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ch  \/  ph ) ).
3:2,?: e2 27416  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ph  \/  ch ) ).
4:1,3,?: e12 27512  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ps  \/  ch ) ).
5:4,?: e2 27416  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ch  \/  ps ) ).
6:5:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ph )  ->  ( ch  \/  ps ) ) ).
7::  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ps ) ).
8:7,?: e2 27416  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ps  \/  ch ) ).
9:1,8,?: e12 27512  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ph  \/  ch ) ).
10:9,?: e2 27416  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ph ) ).
11:10:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ps )  ->  ( ch  \/  ph ) ) ).
12:6,11,?: e11 27473  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ph )  <->  ( ch  \/  ps ) ) ).
qed:12:  |-  ( ( ph  <->  ps )  ->  ( ( ch  \/  ph )  <->  ( ch  \/  ps ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  \/  ph ) 
 <->  ( ch  \/  ps ) ) )
 
Theorembitr3VD 27638 Virtual deduction proof of bitr3 27288. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2:1,?: e1_ 27412  |-  (. ( ph  <->  ps )  ->.  ( ps  <->  ph ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ph  <->  ch ) ).
4:3,?: e2 27416  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ch  <->  ph ) ).
5:2,4,?: e12 27512  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ps  <->  ch ) ).
6:5:  |-  (. ( ph  <->  ps )  ->.  ( ( ph  <->  ch )  ->  ( ps  <->  ch ) ) ).
qed:6:  |-  ( ( ph  <->  ps )  ->  ( ( ph  <->  ch )  ->  ( ps  <->  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  ->  ( ps 
 <->  ch ) ) )
 
Theorem3orbi123VD 27639 Virtual deduction proof of 3orbi123 27289. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) ) ).
2:1,?: e1_ 27412  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ph  <->  ps ) ).
3:1,?: e1_ 27412  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ch  <->  th ) ).
4:1,?: e1_ 27412  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ta  <->  et ) ).
5:2,3,?: e11 27473  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ).
6:5,4,?: e11 27473  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
7:?:  |-  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ph  \/  ch  \/  ta ) )
8:6,7,?: e10 27480  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
9:?:  |-  ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) )
10:8,9,?: e10 27480  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch  \/  ta )  <->  ( ps  \/  th  \/  et ) ) ).
qed:10:  |-  ( ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->  ( ( ph  \/  ch  \/  ta )  <->  ( ps  \/  th  \/  et ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ch 
 <-> 
 th )  /\  ( ta 
 <->  et ) )  ->  ( ( ph  \/  ch 
 \/  ta )  <->  ( ps  \/  th 
 \/  et ) ) )
 
Theoremsbc3orgVD 27640 Virtual deduction proof of sbc3org 27311. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1,?: e1_ 27412  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
3::  |-  ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) )
32:3:  |-  A. x ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) )
33:1,32,?: e10 27480  |-  (. A  e.  B  ->.  [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) ) ).
4:1,33,?: e11 27473  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ).
5:2,4,?: e11 27473  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
6:1,?: e1_ 27412  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) ).
7:6,?: e1_ 27412  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
8:5,7,?: e11 27473  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
9:?:  |-  ( ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) )
10:8,9,?: e10 27480  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) ).
qed:10:  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ]. ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps  \/  [. A  /  x ].
 ch ) ) )
 
Theorem19.21a3con13vVD 27641* Virtual deduction proof of alrim3con13v 27312. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  ->  A. x ph )  ->.  ( ph  ->  A. x ph ) ).
2::  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ( ps  /\  ph  /\  ch ) ).
3:2,?: e2 27416  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ps ).
4:2,?: e2 27416  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ph ).
5:2,?: e2 27416  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ch ).
6:1,4,?: e12 27512  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ph ).
7:3,?: e2 27416  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ps ).
8:5,?: e2 27416  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ch ).
9:7,6,8,?: e222 27421  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ( A. x ps  /\  A. x ph  /\  A. x ch ) ).
10:9,?: e2 27416  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ( ps  /\  ph  /\  ch ) ).
11:10:in2  |-  (. ( ph  ->  A. x ph )  ->.  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps  /\  ph  /\  ch ) ) ).
qed:11:in1  |-  ( ( ph  ->  A. x ph )  ->  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps  /\  ph  /\  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  A. x ph )  ->  ( ( ps  /\  ph  /\  ch )  ->  A. x ( ps 
 /\  ph  /\  ch )
 ) )
 
TheoremexbirVD 27642 Virtual deduction proof of exbir 1361. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ).
2::  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,.  ( ph  /\  ps )  ->.  ( ph  /\  ps ) ).
3::  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,.  ( ph  /\  ps ) ,  th  ->.  th ).
5:1,2,?: e12 27512  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps )  ->.  ( ch  <->  th ) ).
6:3,5,?: e32 27546  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps ) ,  th  ->.  ch ).
7:6:  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps )  ->.  ( th  ->  ch ) ).
8:7:  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->.  ( ( ph  /\  ps )  ->  ( th  ->  ch ) ) ).
9:8,?: e1_ 27412  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->.  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) ) ).
qed:9:  |-  ( ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) ) )
(Contributed by Alan Sare, 13-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) ) )
 
TheoremexbiriVD 27643 Virtual deduction proof of exbiri 608. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )
2::  |-  (. ph  ->.  ph ).
3::  |-  (. ph ,. ps  ->.  ps ).
4::  |-  (. ph ,. ps ,. th  ->.  th ).
5:2,1,?: e10 27480  |-  (. ph  ->.  ( ps  ->  ( ch  <->  th ) ) ).
6:3,5,?: e21 27518  |-  (. ph ,. ps  ->.  ( ch  <->  th ) ).
7:4,6,?: e32 27546  |-  (. ph ,. ps ,. th  ->.  ch ).
8:7:  |-  (. ph ,. ps  ->.  ( th  ->  ch ) ).
9:8:  |-  (. ph  ->.  ( ps  ->  ( th  ->  ch ) ) ).
qed:9:  |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
 
Theoremra4sbc2VD 27644* Virtual deduction proof of ra4sbc2 27313. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  (. A  e.  B ,. C  e.  D  ->.  C  e.  D ).
3::  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x  e.  B A. y  e.  D ph ).
4:1,3,?: e13 27536  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  [. A  /  x ]. A. y  e.  D ph ).
5:1,4,?: e13 27536  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  A. y  e.  D [. A  /  x ]. ph ).
6:2,5,?: e23 27543  |-  (. A  e.  B ,. C  e.  D ,. A. x  e.  B  A. y  e.  D ph  ->.  [. C  /  y ]. [. A  /  x ]. ph ).
7:6:  |-  (. A  e.  B ,. C  e.  D  ->.  ( A. x  e.  B  A. y  e.  D ph  ->  [. C  /  y ]. [. A  /  x ]. ph ) ).
8:7:  |-  (. A  e.  B  ->.  ( C  e.  D  ->  ( A. x  e.  B A. y  e.  D ph  ->  [. C  /  y ]. [. A  /  x ]. ph ) ) ).
qed:8:  |-  ( A  e.  B  ->  ( C  e.  D  ->  ( A. x  e.  B A. y  e.  D ph  ->  [. C  /  y ]. [. A  /  x ]. ph ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  ( C  e.  D  ->  (
 A. x  e.  B  A. y  e.  D  ph  -> 
 [. C  /  y ]. [. A  /  x ].
 ph ) ) )
 
Theorem3impexpVD 27645 Virtual deduction proof of 3impexp 1362. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
2::  |-  ( ( ph  /\  ps  /\  ch )  <->  ( ( ph  /\  ps )  /\  ch ) )
3:1,2,?: e10 27480  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
4:3,?: e1_ 27412  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
5:4,?: e1_ 27412  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
6:5:  |-  ( ( ( ph  /\  ps  /\  ch )  ->  th )  ->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
7::  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) ).
8:7,?: e1_ 27412  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
9:8,?: e1_ 27412  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
10:2,9,?: e01 27476  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
11:10:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  th ) )
qed:6,11,?: e00 27556  |-  ( ( ( ph  /\  ps  /\  ch )  ->  th )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  th )  <->  (
 ph  ->  ( ps  ->  ( ch  ->  th )
 ) ) )
 
Theorem3impexpbicomVD 27646 Virtual deduction proof of 3impexpbicom 1363. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) ) ).
2::  |-  ( ( th  <->  ta )  <->  ( ta  <->  th ) )
3:1,2,?: e10 27480  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
4:3,?: e1_ 27412  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) ).
5:4:  |-  ( ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
6::  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) ).
7:6,?: e1_ 27412  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
8:7,2,?: e10 27480  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) ) ).
9:8:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) ) )
qed:5,9,?: e00 27556  |-  ( ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  ( th 
 <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
 
Theorem3impexpbicomiVD 27647 Virtual deduction proof of 3impexpbicomi 1364. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )
qed:1,?: e0_ 27560  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
 
TheoremsbcoreleleqVD 27648* Virtual deduction proof of sbcoreleleq 27314. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1,?: e1_ 27412  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) ).
3:1,?: e1_ 27412  |-  (. A  e.  B  ->.  ( [. A  /  y ]. y  e.  x  <->  A  e.  x ) ).
4:1,?: e1_ 27412  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  =  y  <->  x  =  A ) ).
5:2,3,4,?: e111 27459  |-  (. A  e.  B  ->.  ( ( x  e.  A  \/  A  e.  x  \/  x  =  A )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ).
6:1,?: e1_ 27412  |-  (. A  e.  B  ->.  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( [. A  /  y ]. x  e.  y  \/  [. A  /  y ]. y  e.  x  \/  [. A  /  y ]. x  =  y ) ) ).
7:5,6: e11 27473  |-  (. A  e.  B  ->.  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) ).
qed:7:  |-  ( A  e.  B  ->  ( [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y )  <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y
 ) 
 <->  ( x  e.  A  \/  A  e.  x  \/  x  =  A )
 ) )
 
Theoremhbra2VD 27649* Virtual deduction proof of nfra2 2568. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  ( A. y  e.  B A. x  e.  A ph  ->  A. y A. y  e.  B A. x  e.  A ph )
2::  |-  ( A. x  e.  A A. y  e.  B ph  <->  A. y  e.  B A. x  e.  A ph )
3:1,2,?: e00 27556  |-  ( A. x  e.  A A. y  e.  B ph  ->  A. y A. y  e.  B A. x  e.  A ph )
4:2:  |-  A. y ( A. x  e.  A A. y  e.  B ph  <->  A. y  e.  B A. x  e.  A ph )
5:4,?: e0_ 27560  |-  ( A. y A. x  e.  A A. y  e.  B ph  <->  A. y A. y  e.  B A. x  e.  A ph )
qed:3,5,?: e00 27556  |-  ( A. x  e.  A A. y  e.  B ph  ->  A. y A. x  e.  A A. y  e.  B ph )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  A. y  e.  B  ph  ->  A. y A. x  e.  A  A. y  e.  B  ph )
 
TheoremtratrbVD 27650* Virtual deduction proof of tratrb 27315. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  ( Tr  A  /\  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ).
2:1,?: e1_ 27412  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  Tr  A ).
3:1,?: e1_ 27412  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
4:1,?: e1_ 27412  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  B  e.  A ).
5::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( x  e.  y  /\  y  e.  B ) ).
6:5,?: e2 27416  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  x  e.  y ).
7:5,?: e2 27416  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  y  e.  B ).
8:2,7,4,?: e121 27441  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  y  e.  A ).
9:2,6,8,?: e122 27438  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  x  e.  A ).
10::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  B  e.  x  ->.  B  e.  x ).
11:6,7,10,?: e223 27420  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  B  e.  x  ->.  ( x  e.  y  /\  y  e.  B  /\  B  e.  x ) ).
12:11:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( B  e.  x  ->  ( x  e.  y  /\  y  e.  B  /\  B  e.  x ) ) ).
13::  |-  -.  ( x  e.  y  /\  y  e.  B  /\  B  e.  x )
14:12,13,?: e20 27515  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  -.  B  e.  x ).
15::  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  x  =  B  ->.  x  =  B ).
16:7,15,?: e23 27543  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  x  =  B  ->.  y  e.  x ).
17:6,16,?: e23 27543  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B ) ,  x  =  B  ->.  ( x  e.  y  /\  y  e.  x ) ).
18:17:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( x  =  B  ->  ( x  e.  y  /\  y  e.  x ) ) ).
19::  |-  -.  ( x  e.  y  /\  y  e.  x )
20:18,19,?: e20 27515  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  -.  x  =  B ).
21:3,?: e1_ 27412  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. y  e.  A  A. x  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
22:21,9,4,?: e121 27441  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  [. x  /  x ]. [. B  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
23:22,?: e2 27416  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  [. B  /  y ]. ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
24:4,23,?: e12 27512  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  ( x  e.  B  \/  B  e.  x  \/  x  =  B ) ).
25:14,20,24,?: e222 27421  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) ,  ( x  e.  y  /\  y  e.  B )  ->.  x  e.  B ).
26:25:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  ( ( x  e.  y  /\  y  e.  B )  ->  x  e.  B ) ).
27::  |-  ( A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  ->  A. y A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
28:27,?: e0_ 27560  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->  A. y ( Tr  A  /\  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) )
29:28,26:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. y ( ( x  e.  y  /\  y  e.  B )  ->  x  e.  B ) ).
30::  |-  ( A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  ->  A. x A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
31:30,?: e0_ 27560  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->  A. x ( Tr  A  /\  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) )
32:31,29:  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  A. x  A. y ( ( x  e.  y  /\  y  e.  B )  ->  x  e.  B ) ).
33:32,?: e1_ 27412  |-  (. ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->.  Tr  B ).
qed:33:  |-  ( ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A )  ->  Tr  B )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  y  e.  x  \/  x  =  y )  /\  B  e.  A ) 
 ->  Tr  B )
 
Theorem3ax5VD 27651 Virtual deduction proof of 3ax5 27316. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. A. x ( ph  ->  ( ps  ->  ch ) )  ->.  A. x ( ph  ->  ( ps  ->  ch ) ) ).
2:1,?: e1_ 27412  |-  (. A. x ( ph  ->  ( ps  ->  ch ) )  ->.  ( A. x ph  ->  A. x ( ps  ->  ch ) ) ).
3::  |-  ( A. x ( ps  ->  ch )  ->  ( A. x ps  ->  A. x ch ) )
4:2,3,?: e10 27480  |-  (. A. x ( ph  ->  ( ps  ->  ch ) )  ->.  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) ).
qed:4:  |-  ( A. x ( ph  ->  ( ps  ->  ch ) )  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x ( ph  ->  ( ps  ->  ch )
 )  ->  ( A. x ph  ->  ( A. x ps  ->  A. x ch ) ) )
 
Theoremsyl5impVD 27652 Virtual deduction proof of syl5imp 27290. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  (. ( ph  ->  ( ps  ->  ch ) )  ->.  ( ph  ->  ( ps  ->  ch ) ) ).
2:1,?: e1_ 27412  |-  (. ( ph  ->  ( ps  ->  ch ) )  ->.  ( ps  ->  ( ph  ->  ch ) ) ).
3::  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( th  ->  ps ) ).
4:3,2,?: e21 27518  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( th  ->  ( ph  ->  ch ) ) ).
5:4,?: e2 27416  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( ph  ->  ( th  ->  ch ) ) ).
6:5:  |-  (. ( ph  ->  ( ps  ->  ch ) )  ->.  ( ( th  ->  ps )  ->  ( ph  ->  ( th  ->  ch ) ) ) ).
qed:6:  |-  ( ( ph  ->  ( ps  ->  ch ) )  ->  ( ( th  ->  ps )  ->  ( ph  ->  ( th  ->  ch ) ) ) )
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( th  ->  ps )  ->  ( ph  ->  ( th  ->  ch )
 ) ) )
 
TheoremidiVD 27653 Virtual deduction proof of idi 2. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1::  |-  ph
qed:1,?: e0_ 27560  |-  ph
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   =>    |-  ph
 
TheoremancomsimpVD 27654 Closed form of ancoms 441. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::  |-  ( ( ph  /\  ps )  <->  ( ps  /\  ph ) )
qed:1,?: e0_ 27560  |-  ( ( ( ph  /\  ps )  ->  ch )  <->  ( ( ps  /\  ph )  ->  ch ) )
(Contributed by Alan Sare, 25-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  /\  ps )  ->  ch )  <->  ( ( ps 
 /\  ph )  ->  ch )
 )
 
Theoremssralv2VD 27655* Quantification restricted to a subclass for two quantifiers. ssralv 3179 for two quantifiers. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ssralv2 27310 is ssralv2VD 27655 without virtual deductions and was automatically derived from ssralv2VD 27655.
1::  |-  (. ( A  C_  B  /\  C  C_  D )  ->.  ( A  C_  B  /\  C  C_  D ) ).
2::  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x  e.  B A. y  e.  D ph ).
3:1:  |-  (. ( A  C_  B  /\  C  C_  D )  ->.  A  C_  B ).
4:3,2:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x  e.  A A. y  e.  D ph ).
5:4:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x ( x  e.  A  ->  A. y  e.  D ph ) ).
6:5:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  ( x  e.  A  ->  A. y  e.  D ph ) ).
7::  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph ,  x  e.  A  ->.  x  e.  A ).
8:7,6:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph ,  x  e.  A  ->.  A. y  e.  D ph ).
9:1:  |-  (. ( A  C_  B  /\  C  C_  D )  ->.  C  C_  D ).
10:9,8:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph ,  x  e.  A  ->.  A. y  e.  C ph ).
11:10:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  ( x  e.  A  ->  A. y  e.  C ph ) ).
12::  |-  ( ( A  C_  B  /\  C  C_  D )  ->  A. x ( A  C_  B  /\  C  C_  D ) )
13::  |-  ( A. x  e.  B A. y  e.  D ph  ->  A. x A. x  e.  B A. y  e.  D ph )
14:12,13,11:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x ( x  e.  A  ->  A. y  e.  C ph ) ).
15:14:  |-  (. ( A  C_  B  /\  C  C_  D ) ,. A. x  e.  B  A. y  e.  D ph  ->.  A. x  e.  A A. y  e.  C ph ).
16:15:  |-  (. ( A  C_  B  /\  C  C_  D )  ->.  ( A. x  e.  B A. y  e.  D ph  ->  A. x  e.  A A. y  e.  C ph ) ).
qed:16:  |-  ( ( A  C_  B  /\  C  C_  D )  ->  ( A. x  e.  B A. y  e.  D ph  ->  A. x  e.  A A. y  e.  C ph ) )
(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  C_  B  /\  C  C_  D )  ->  ( A. x  e.  B  A. y  e.  D  ph  ->  A. x  e.  A  A. y  e.  C  ph ) )
 
TheoremordelordALTVD 27656 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4351 using the Axiom of Regularity indirectly through dford2 7254. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that  _E  Fr  A because this is inferred by the Axiom of Regularity. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ordelordALT 27317 is ordelordALTVD 27656 without virtual deductions and was automatically derived from ordelordALTVD 27656 using the tools program translate..without..overwriting.cmd and Metamath's minimize command.
1::  |-  (. ( Ord  A  /\  B  e.  A )  ->.  ( Ord  A  /\  B  e.  A ) ).
2:1:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  Ord  A ).
3:1:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  B  e.  A ).
4:2:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  Tr  A ).
5:2:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ).
6:4,3:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  B  C_  A ).
7:6,6,5:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  A. x  e.  B  A. y  e.  B ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ).
8::  |-  ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
9:8:  |-  A. y ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
10:9:  |-  A. y  e.  A ( ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
11:10:  |-  ( A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
12:11:  |-  A. x ( A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
13:12:  |-  A. x  e.  A ( A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
14:13:  |-  ( A. x  e.  A A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x )  <->  A. x  e.  A A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) )
15:14,5:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  y  e.  x  \/  x  =  y ) ).
16:4,15,3:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  Tr  B ).
17:16,7:  |-  (. ( Ord  A  /\  B  e.  A )  ->.  Ord  B ).
qed:17:  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( Ord  A  /\  B  e.  A )  ->  Ord  B )
 
TheoremequncomVD 27657 If a class equals the union of two other classes, then it equals the union of those two classes commuted. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncom 3262 is equncomVD 27657 without virtual deductions and was automatically derived from equncomVD 27657.
1::  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( B  u.  C ) ).
2::  |-  ( B  u.  C )  =  ( C  u.  B )
3:1,2:  |-  (. A  =  ( B  u.  C )  ->.  A  =  ( C  u.  B ) ).
4:3:  |-  ( A  =  ( B  u.  C )  ->  A  =  ( C  u.  B ) )
5::  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( C  u.  B ) ).
6:5,2:  |-  (. A  =  ( C  u.  B )  ->.  A  =  ( B  u.  C ) ).
7:6:  |-  ( A  =  ( C  u.  B )  ->  A  =  ( B  u.  C ) )
8:4,7:  |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B ) )
(Contributed by Alan Sare, 17-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B ) )
 
TheoremequncomiVD 27658 Inference form of equncom 3262. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. equncomi 3263 is equncomiVD 27658 without virtual deductions and was automatically derived from equncomiVD 27658.
h1::  |-  A  =  ( B  u.  C )
qed:1:  |-  A  =  ( C  u.  B )
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  =  ( B  u.  C )   =>    |-  A  =  ( C  u.  B )
 
TheoremsucidALTVD 27659 A set belongs to its successor. Alternate proof of sucid 4408. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucidALT 27660 is sucidALTVD 27659 without virtual deductions and was automatically derived from sucidALTVD 27659. This proof illustrates that completeusersproof.cmd will generate a Metamath proof from any User's Proof which is "conventional" in the sense that no step is a virtual deduction, provided that all necessary unification theorems and transformation deductions are in set.mm. completeusersproof.cmd automatically converts such a conventional proof into a Virtual Deduction proof for which each step happens to be a 0-virtual hypothesis virtual deduction. The user does not need to search for reference theorem labels or deduction labels nor does he(she) need to use theorems and deductions which unify with reference theorems and deductions in set.mm. All that is necessary is that each theorem or deduction of the User's Proof unifies with some reference theorem or deduction in set.mm or is a semantic variation of some theorem or deduction which unifies with some reference theorem or deduction in set.mm. The definition of "semantic variation" has not been precisely defined. If it is obvious that a theorem or deduction has the same meaning as another theorem or deduction, then it is a semantic variation of the latter theorem or deduction. For example, step 4 of the User's Proof is a semantic variation of the definition (axiom)  suc  A  =  ( A  u.  { A } ), which unifies with df-suc 4335, a reference definition (axiom) in set.mm. Also, a theorem or deduction is said to be a semantic variation of a another theorem or deduction if it is obvious upon cursory inspection that it has the same meaning as a weaker form of the latter theorem or deduction. For example, the deduction  Ord  A infers  A. x  e.  A A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x ) is a semantic variation of the theorem  ( Ord  A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ) ), which unifies with the set.mm reference definition (axiom) dford2 7254.
h1::  |-  A  e.  _V
2:1:  |-  A  e.  { A }
3:2:  |-  A  e.  ( { A }  u.  A )
4::  |-  suc  A  =  ( { A }  u.  A )
qed:3,4:  |-  A  e.  suc  A
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  A  e.  suc  A
 
TheoremsucidALT 27660 A set belongs to its successor. This proof was automatically derived from sucidALTVD 27659 using translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  A  e.  suc  A
 
TheoremsucidVD 27661 A set belongs to its successor. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucid 4408 is sucidVD 27661 without virtual deductions and was automatically derived from sucidVD 27661.
h1::  |-  A  e.  _V
2:1:  |-  A  e.  { A }
3:2:  |-  A  e.  ( A  u.  { A } )
4::  |-  suc  A  =  ( A  u.  { A } )
qed:3,4:  |-  A  e.  suc  A
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  A  e.  suc  A
 
Theoremimbi12VD 27662 Implication form of imbi12i 318. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi12 27298 is imbi12VD 27662 without virtual deductions and was automatically derived from imbi12VD 27662.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ch  <->  th ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ph  ->  ch )  ->.  ( ph  ->  ch ) ).
4:1,3:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ph  ->  ch )  ->.  ( ps  ->  ch ) ).
5:2,4:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ph  ->  ch )  ->.  ( ps  ->  th ) ).
6:5:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ph  ->  ch )  ->  ( ps  ->  th ) ) ).
7::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ps  ->  th )  ->.  ( ps  ->  th ) ).
8:1,7:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ps  ->  th )  ->.  ( ph  ->  th ) ).
9:2,8:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ps  ->  th )  ->.  ( ph  ->  ch ) ).
10:9:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ps  ->  th )  ->  ( ph  ->  ch ) ) ).
11:6,10:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ).
12:11:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) ).
qed:12:  |-  ( ( ph  <->  ps )  ->  ( ( ch  <->  th )  ->  ( ( ph  ->  ch )  <->  ( ps  ->  th ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  <->  th )  ->  (
 ( ph  ->  ch )  <->  ( ps  ->  th )
 ) ) )
 
Theoremimbi13VD 27663 Join three logical equivalences to form equivalence of implications. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi13 27299 is imbi13VD 27663 without virtual deductions and was automatically derived from imbi13VD 27663.
1::  |-  (. ( ph  <->  ps )  ->.  ( ph  <->  ps ) ).
2::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ch  <->  th ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ta  <->  et )  ->.  ( ta  <->  et ) ).
4:2,3:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ta  <->  et )  ->.  ( ( ch  ->  ta )  <->  ( th  ->  et ) ) ).
5:1,4:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th ) ,. ( ta  <->  et )  ->.  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ).
6:5:  |-  (. ( ph  <->  ps ) ,. ( ch  <->  th )  ->.  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ).
7:6:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ) ).
qed:7:  |-  ( ( ph  <->  ps )  ->  ( ( ch  <->  th )  ->  ( ( ta  <->  et )  ->  ( ( ph  ->  ( ch  ->  ta ) )  <->  ( ps  ->  ( th  ->  et ) ) ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ch  <->  th )  ->  (
 ( ta  <->  et )  ->  (
 ( ph  ->  ( ch 
 ->  ta ) )  <->  ( ps  ->  ( th  ->  et )
 ) ) ) ) )
 
Theoremsbcim2gVD 27664 Distribution of class substitution over a left-nested implication. Similar to sbcimg 2976. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcim2g 27318 is sbcim2gVD 27664 without virtual deductions and was automatically derived from sbcim2gVD 27664.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ).
3:1,2:  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ).
4:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ps  ->  ch )  <->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
5:3,4:  |-  (. A  e.  B ,. [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->.  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
6:5:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  ->  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) ).
7::  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ).
8:4,7:  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ).
9:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ( ps  ->  ch ) ) ) ).
10:8,9:  |-  (. A  e.  B ,. ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->.  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ).
11:10:  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) )  ->  [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) ) ) ).
12:6,11:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) ).
qed:12:  |-  ( A  e.  B  ->  ( [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch ) ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. ( ph  ->  ( ps  ->  ch ) )  <->  ( [. A  /  x ]. ph  ->  (
 [. A  /  x ].
 ps  ->  [. A  /  x ].
 ch ) ) ) )
 
TheoremsbcbiVD 27665 Implication form of sbcbiiOLD 2991. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcbi 27319 is sbcbiVD 27665 without virtual deductions and was automatically derived from sbcbiVD 27665.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  (. A  e.  B ,. A. x ( ph  <->  ps )  ->.  A. x ( ph  <->  ps ) ).
3:1,2:  |-  (. A  e.  B ,. A. x ( ph  <->  ps )  ->.  [. A  /  x ]. ( ph  <->  ps ) ).
4:1,3:  |-  (. A  e.  B ,. A. x ( ph  <->  ps )  ->.  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ).
5:4:  |-  (. A  e.  B  ->.  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) ).
qed:5:  |-  ( A  e.  B  ->  ( A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ]. ps ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 A. x ( ph  <->  ps )  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps ) ) )
 
TheoremtrsbcVD 27666* Formula-building inference rule for class substitution, substituting a class variable for the set variable of the transitivity predicate. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trsbc 27320 is trsbcVD 27666 without virtual deductions and was automatically derived from trsbcVD 27666.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. z  e.  y  <->  z  e.  y ) ).
3:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  x  <->  y  e.  A ) ).
4:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. z  e.  x  <->  z  e.  A ) ).
5:1,2,3,4:  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. z  e.  y  ->  ( [. A  /  x ]. y  e.  x  ->  [. A  /  x ]. z  e.  x ) )  <->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) ) ).
6:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( [. A  /  x ]. z  e.  y  ->  ( [. A  /  x ]. y  e.  x  ->  [. A  /  x ]. z  e.  x ) ) ) ).
7:5,6:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) ) ) ).
8::  |-  ( ( z  e.  y  ->  ( y  e.  A  ->  z  e.  A ) )  <->  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
9:7,8:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
10::  |-  ( ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
11:10:  |-  A. x ( ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
12:1,11:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( z  e.  y  ->  ( y  e.  x  ->  z  e.  x ) )  <->  [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) ) ).
13:9,12:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
14:13:  |-  (. A  e.  B  ->.  A. y ( [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
15:14:  |-  (. A  e.  B  ->.  ( A. y [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
16:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. y [. A  /  x ]. ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) ) ).
17:15,16:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
18:17:  |-  (. A  e.  B  ->.  A. z ( [. A  /  x ]. A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
19:18:  |-  (. A  e.  B  ->.  ( A. z [. A  /  x ]. A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
20:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. z A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. z [. A  /  x ]. A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) ) ).
21:19,20:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. z A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) ) ).
22::  |-  ( Tr  A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  A )  ->  z  e.  A ) )
23:21,22:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. z A. y ( (  z  e.  y  /\  y  e.  x )  ->  z  e.  x )  <->  Tr  A ) ).
24::  |-  ( Tr  x  <->  A. z A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
25:24:  |-  A. x ( Tr  x  <->  A. z A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) )
26:1,25:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. Tr  x  <->  [. A  /  x ]. A. z A. y ( ( z  e.  y  /\  y  e.  x )  ->  z  e.  x ) ) ).
27:23,26:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. Tr  x  <->  Tr  A ) ).
qed:27:  |-  ( A  e.  B  ->  ( [. A  /  x ]. Tr  x  <->  Tr  A ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ].
 Tr  x  <->  Tr  A ) )
 
TheoremtruniALTVD 27667* The union of a class of transitive sets is transitive. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. truniALT 27321 is truniALTVD 27667 without virtual deductions and was automatically derived from truniALTVD 27667.
1::  |-  (. A. x  e.  A Tr  x  ->.  A. x  e.  A  Tr  x ).
2::  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  ( z  e.  y  /\  y  e.  U. A ) ).
3:2:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  z  e.  y ).
4:2:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  y  e.  U. A ).
5:4:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  E. q ( y  e.  q  /\  q  e.  A ) ).
6::  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  ( y  e.  q  /\  q  e.  A ) ).
7:6:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  y  e.  q ).
8:6:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  q  e.  A ).
9:1,8:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  [ q  /  x ] Tr  x ).
10:8,9:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  Tr  q ).
11:3,7,10:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  z  e.  q ).
12:11,8:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A ) ,  ( y  e.  q  /\  q  e.  A )  ->.  z  e.  U. A ).
13:12:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  ( ( y  e.  q  /\  q  e.  A )  ->  z  e.  U. A ) ).
14:13:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  A. q ( ( y  e.  q  /\  q  e.  A )  ->  z  e.  U. A ) ).
15:14:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  ( E. q ( y  e.  q  /\  q  e.  A )  ->  z  e.  U. A ) ).
16:5,15:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  U. A )  ->.  z  e.  U. A ).
17:16:  |-  (. A. x  e.  A Tr  x  ->.  ( ( z  e.  y  /\  y  e.  U. A )  ->  z  e.  U. A ) ).
18:17:  |-  (. A. x  e.  A Tr  x  ->.  A. z A. y ( ( z  e.  y  /\  y  e.  U. A )  ->  z  e.  U. A ) ).
19:18:  |-  (. A. x  e.  A Tr  x  ->.  Tr  U. A ).
qed:19:  |-  ( A. x  e.  A Tr  x  ->  Tr  U. A )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  Tr  x  ->  Tr  U. A )
 
Theoremee33VD 27668 Non-virtual deduction form of e33 27522. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ee33 27300 is ee33VD 27668 without virtual deductions and was automatically derived from ee33VD 27668.
h1::  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
h2::  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
h3::  |-  ( th  ->  ( ta  ->  et ) )
4:1,3:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  ->  et ) ) ) )
5:4:  |-  ( ta  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
6:2,5:  |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) ) )
7:6:  |-  ( ps  ->  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) ) )
8:7:  |-  ( ch  ->  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) ) )
qed:8:  |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  et ) ) )
 
TheoremtrintALTVD 27669* The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 27670. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trintALT 27670 is trintALTVD 27669 without virtual deductions and was automatically derived from trintALTVD 27669.
1::  |-  (. A. x  e.  A Tr  x  ->.  A. x  e.  A Tr  x ).
2::  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  ( z  e.  y  /\  y  e.  |^| A ) ).
3:2:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  z  e.  y ).
4:2:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  y  e.  |^| A ).
5:4:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  A. q  e.  A y  e.  q ).
6:5:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  ( q  e.  A  ->  y  e.  q ) ).
7::  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  q  e.  A ).
8:7,6:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  y  e.  q ).
9:7,1:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  [ q  /  x ] Tr  x ).
10:7,9:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  Tr  q ).
11:10,3,8:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A ) ,  q  e.  A  ->.  z  e.  q ).
12:11:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  ( q  e.  A  ->  z  e.  q ) ).
13:12:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  A. q ( q  e.  A  ->  z  e.  q ) ).
14:13:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  A. q  e.  A z  e.  q ).
15:3,14:  |-  (. A. x  e.  A Tr  x ,. ( z  e.  y  /\  y  e.  |^| A )  ->.  z  e.  |^| A ).
16:15:  |-  (. A. x  e.  A Tr  x  ->.  ( ( z  e.  y  /\  y  e.  |^| A )  ->  z  e.  |^| A ) ).
17:16:  |-  (. A. x  e.  A Tr  x  ->.  A. z A. y ( ( z  e.  y  /\  y  e.  |^| A )  ->  z  e.  |^| A ) ).
18:17:  |-  (. A. x  e.  A Tr  x  ->.  Tr  |^| A ).
qed:18:  |-  ( A. x  e.  A Tr  x  ->  Tr  |^| A )
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
 
TheoremtrintALT 27670* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 27670 is an alternative proof of trint 4068. trintALT 27670 is trintALTVD 27669 without virtual deductions and was automatically derived from trintALTVD 27669 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
 
Theoremundif3VD 27671 The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 3371. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. undif3 3371 is undif3VD 27671 without virtual deductions and was automatically derived from undif3VD 27671.
1::  |-  ( x  e.  ( A  u.  ( B  \  C ) )  <->  ( x  e.  A  \/  x  e.  ( B  \  C ) ) )
2::  |-  ( x  e.  ( B  \  C )  <->  ( x  e.  B  /\  -.  x  e.  C ) )
3:2:  |-  ( ( x  e.  A  \/  x  e.  ( B  \  C ) )  <->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
4:1,3:  |-  ( x  e.  ( A  u.  ( B  \  C ) )  <->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
5::  |-  (. x  e.  A  ->.  x  e.  A ).
6:5:  |-  (. x  e.  A  ->.  ( x  e.  A  \/  x  e.  B ) ).
7:5:  |-  (. x  e.  A  ->.  ( -.  x  e.  C  \/  x  e.  A ) ).
8:6,7:  |-  (. x  e.  A  ->.  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) ).
9:8:  |-  ( x  e.  A  ->  ( ( x  e.  A  \/  x  e.  B )  /\  (  -.  x  e.  C  \/  x  e.  A ) ) )
10::  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( x  e.  B  /\  -.  x  e.  C ) ).
11:10:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  x  e.  B ).
12:10:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  -.  x  e.  C  ).
13:11:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( x  e.  A  \/  x  e.  B ) ).
14:12:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( -.  x  e.  C  \/  x  e.  A ) ).
15:13,14:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) ).
16:15:  |-  ( ( x  e.  B  /\  -.  x  e.  C )  ->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
17:9,16:  |-  ( ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) )  ->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
18::  |-  (. ( x  e.  A  /\  -.  x  e.  C )  ->.  ( x  e.  A  /\  -.  x  e.  C ) ).
19:18:  |-  (. ( x  e.  A  /\  -.  x  e.  C )  ->.  x  e.  A ).
20:18:  |-  (. ( x  e.  A  /\  -.  x  e.  C )  ->.  -.  x  e.  C  ).
21:18:  |-  (. ( x  e.  A  /\  -.  x  e.  C )  ->.  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) ).
22:21:  |-  ( ( x  e.  A  /\  -.  x  e.  C )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
23::  |-  (. ( x  e.  A  /\  x  e.  A )  ->.  ( x  e.  A  /\  x  e.  A ) ).
24:23:  |-  (. ( x  e.  A  /\  x  e.  A )  ->.  x  e.  A ).
25:24:  |-  (. ( x  e.  A  /\  x  e.  A )  ->.  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) ).
26:25:  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( x  e.  A  \/  (  x  e.  B  /\  -.  x  e.  C ) ) )
27:10:  |-  (. ( x  e.  B  /\  -.  x  e.  C )  ->.  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) ).
28:27:  |-  ( ( x  e.  B  /\  -.  x  e.  C )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
29::  |-  (. ( x  e.  B  /\  x  e.  A )  ->.  ( x  e.  B  /\  x  e.  A ) ).
30:29:  |-  (. ( x  e.  B  /\  x  e.  A )  ->.  x  e.  A ).
31:30:  |-  (. ( x  e.  B  /\  x  e.  A )  ->.  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) ).
32:31:  |-  ( ( x  e.  B  /\  x  e.  A )  ->  ( x  e.  A  \/  (  x  e.  B  /\  -.  x  e.  C ) ) )
33:22,26:  |-  ( ( ( x  e.  A  /\  -.  x  e.  C )  \/  ( x  e.  A  /\  x  e.  A ) )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
34:28,32:  |-  ( ( ( x  e.  B  /\  -.  x  e.  C )  \/  ( x  e.  B  /\  x  e.  A ) )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
35:33,34:  |-  ( ( ( ( x  e.  A  /\  -.  x  e.  C )  \/  ( x  e.  A  /\  x  e.  A ) )  \/  ( ( x  e.  B  /\  -.  x  e.  C )  \/  ( x  e.  B  /\  x  e.  A ) ) )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
36::  |-  ( ( ( ( x  e.  A  /\  -.  x  e.  C )  \/  ( x  e.  A  /\  x  e.  A ) )  \/  ( ( x  e.  B  /\  -.  x  e.  C )  \/  ( x  e.  B  /\  x  e.  A ) ) )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
37:36,35:  |-  ( ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) )  ->  ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) ) )
38:17,37:  |-  ( ( x  e.  A  \/  ( x  e.  B  /\  -.  x  e.  C ) )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
39::  |-  ( x  e.  ( C  \  A )  <->  ( x  e.  C  /\  -.  x  e.  A ) )
40:39:  |-  ( -.  x  e.  ( C  \  A )  <->  -.  ( x  e.  C  /\  -.  x  e.  A ) )
41::  |-  ( -.  ( x  e.  C  /\  -.  x  e.  A )  <->  ( -.  x  e.  C  \/  x  e.  A ) )
42:40,41:  |-  ( -.  x  e.  ( C  \  A )  <->  ( -.  x  e.  C  \/  x  e.  A ) )
43::  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B  ) )
44:43,42:  |-  ( ( x  e.  ( A  u.  B )  /\  -.  x  e.  ( C  \  A )  )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  /\  x  e.  A ) ) )
45::  |-  ( x  e.  ( ( A  u.  B )  \  ( C  \  A ) )  <->  (  x  e.  ( A  u.  B )  /\  -.  x  e.  ( C  \  A ) ) )
46:45,44:  |-  ( x  e.  ( ( A  u.  B )  \  ( C  \  A ) )  <->  (  ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
47:4,38:  |-  ( x  e.  ( A  u.  ( B  \  C ) )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ( -.  x  e.  C  \/  x  e.  A ) ) )
48:46,47:  |-  ( x  e.  ( A  u.  ( B  \  C ) )  <->  x  e.  ( ( A  u.  B )  \  ( C  \  A ) ) )
49:48:  |-  A. x ( x  e.  ( A  u.  ( B  \  C ) )  <->  x  e.  ( ( A  u.  B )  \  ( C  \  A ) ) )
qed:49:  |-  ( A  u.  ( B  \  C ) )  =  ( ( A  u.  B )  \  ( C  \  A ) )
(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  u.  ( B  \  C ) )  =  ( ( A  u.  B )  \  ( C 
 \  A ) )
 
TheoremsbcssVD 27672 Virtual deduction proof of sbcss 27322. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcss 27322 is sbcssVD 27672 without virtual deductions and was automatically derived from sbcssVD 27672.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C ) ).
3:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  D  <->  y  e.  [_ A  /  x ]_ D ) ).
4:2,3:  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. y  e.  C  ->  [. A  /  x ]. y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D  ) ) ).
5:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  ( [. A  /  x ]. y  e.  C  ->  [. A  /  x ]. y  e.  D ) ) ).
6:4,5:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) ).
7:6:  |-  (. A  e.  B  ->.  A. y ( [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) ).
8:7:  |-  (. A  e.  B  ->.  ( A. y [. A  /  x ]. ( y  e.  C  ->  y  e.  D )  <->  A. y ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D )  ) ).
9:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. y ( y  e.  C  ->  y  e.  D )  <->  A. y [. A  /  x ]. ( y  e.  C  ->  y  e.  D ) ) ).
10:8,9:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. A. y ( y  e.  C  ->  y  e.  D )  <->  A. y ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D )  ) ).
11::  |-  ( C  C_  D  <->  A. y ( y  e.  C  ->  y  e.  D ) )
110:11:  |-  A. x ( C  C_  D  <->  A. y ( y  e.  C  ->  y  e.  D ) )
12:1,110:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  C_  D  <->  [. A  /  x ]. A. y ( y  e.  C  ->  y  e.  D ) ) ).
13:10,12:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  C_  D  <->  A. y ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) ) ).
14::  |-  ( [_ A  /  x ]_ C  C_  [_ A  /  x ]_ D  <->  A.  y ( y  e.  [_ A  /  x ]_ C  ->  y  e.  [_ A  /  x ]_ D ) )
15:13,14:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. C  C_  D  <->  [_ A  /  x ]_ C  C_  [_ A  /  x ]_ D ) ).
qed:15:  |-  ( A  e.  B  ->  ( [. A  /  x ]. C  C_  D  <->  [_  A  /  x ]_ C  C_  [_ A  /  x ]_ D ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  (
 [. A  /  x ]. C  C_  D  <->  [_ A  /  x ]_ C  C_  [_ A  /  x ]_ D ) )
 
TheoremcsbingVD 27673 Virtual deduction proof of csbing 3318. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbing 3318 is csbingVD 27673 without virtual deductions and was automatically derived from csbingVD 27673.
1::  |-  (. A  e.  B  ->.  A  e.  B ).
2::  |-  ( C  i^i  D )  =  { y  |  ( y  e.  C  /\  y  e.  D )  }
20:2:  |-  A. x ( C  i^i  D )  =  { y  |  ( y  e.  C  /\  y  e.  D ) }
30:1,20:  |-  (. A  e.  B  ->.  [. A  /  x ]. ( C  i^i  D )  =  { y  |  ( y  e.  C  /\  y  e.  D ) } ).
3:1,30:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ ( C  i^i  D )  =  [_ A  /  x ]_ { y  |  ( y  e.  C  /\  y  e.  D ) } ).
4:1:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ { y  |  ( y  e.  C  /\  y  e.  D ) }  =  { y  |  [. A  /  x ]. ( y  e.  C  /\  y  e.  D ) } ).
5:3,4:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ ( C  i^i  D )  =  { y  |  [. A  /  x ]. ( y  e.  C  /\  y  e.  D ) } ).
6:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  C  <->  y  e.  [_ A  /  x ]_ C ) ).
7:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. y  e.  D  <->  y  e.  [_ A  /  x ]_ D ) ).
8:6,7:  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. y  e.  C  /\  [. A  /  x ]. y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D )  ) ).
9:1:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( y  e.  C  /\  y  e.  D )  <->  ( [. A  /  x ]. y  e.  C  /\  [. A  /  x ]. y  e.  D ) ) ).
10:9,8:  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( y  e.  C  /\  y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) ) ).
11:10:  |-  (. A  e.  B  ->.  A. y ( [. A  /  x ]. ( y  e.  C  /\  y  e.  D )  <->  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) ) ).
12:11:  |-  (. A  e.  B  ->.  { y  |  [. A  /  x ]. ( y  e.  C  /\  y  e.  D ) }  =  { y  |  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) } ).
13:5,12:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ ( C  i^i  D )  =  { y  |  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) } ).
14::  |-  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D )  =  {  y  |  ( y  e.  [_ A  /  x ]_ C  /\  y  e.  [_ A  /  x ]_ D ) }
15:13,14:  |-  (. A  e.  B  ->.  [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) ).
qed:15:  |-  ( A  e.  B  ->  [_ A  /  x ]_ ( C  i^i  D )  =  (  [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
 
TheoremonfrALTlem5VD 27674* Virtual deduction proof of onfrALTlem5 27323. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem5 27323 is onfrALTlem5VD 27674 without virtual deductions and was automatically derived from onfrALTlem5VD 27674.
1::  |-  a  e.  _V
2:1:  |-  ( a  i^i  x )  e.  _V
3:2:  |-  ( [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) )
4:3:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  -.  ( a  i^i  x )  =  (/) )
5::  |-  ( ( a  i^i  x )  =/=  (/)  <->  -.  ( a  i^i  x  )  =  (/) )
6:4,5:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =/=  (/) )
7:2:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  /  b ]. -.  b  =  (/) )
8::  |-  ( b  =/=  (/)  <->  -.  b  =  (/) )
9:8:  |-  A. b ( b  =/=  (/)  <->  -.  b  =  (/) )
10:2,9:  |-  ( [. ( a  i^i  x )  /  b ]. b  =/=  (/)  <->  [. ( a  i^i  x )  /  b ]. -.  b  =  (/) )
11:7,10:  |-  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  /  b ]. b  =/=  (/) )
12:6,11:  |-  ( [. ( a  i^i  x )  /  b ]. b  =/=  (/)  <->  (  a  i^i  x )  =/=  (/) )
13:2:  |-  ( [. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x  )  <->  ( a  i^i  x )  C_  ( a  i^i  x ) )
14:12,13:  |-  ( ( [. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x )  /\  [. ( a  i^i  x )  /  b ]. b  =/=  (/) )  <->  ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) ) )
15:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  <->  ( [. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x )  /\  [. ( a  i^i  x )  /  b ]. b  =/=  (/) ) )
16:15,14:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  <->  ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) ) )
17:2:  |-  [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  (  [_ ( a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )
18:2:  |-  [_ ( a  i^i  x )  /  b ]_ b  =  ( a  i^i  x )
19:2:  |-  [_ ( a  i^i  x )  /  b ]_ y  =  y
20:18,19:  |-  ( [_ ( a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )  =  ( ( a  i^i  x )  i^i  y )
21:17,20:  |-  [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  ( (  a  i^i  x )  i^i  y )
22:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/)  <->  [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x )  /  b ]_  (/) )
23:2:  |-  [_ ( a  i^i  x )  /  b ]_ (/)  =  (/)
24:21,23:  |-  ( [_ ( a  i^i  x )  /  b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x )  /  b ]_ (/)  <->  ( ( a  i^i  x )  i^i  y )  =  (/) )
25:22,24:  |-  ( [. ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/)  <->  ( ( a  i^i  x )  i^i  y )  =  (/) )
26:2:  |-  ( [. ( a  i^i  x )  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x ) )
27:25,26:  |-  ( ( [. ( a  i^i  x )  /  b ]. y  e.  b  /\  [.  ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/) )  <->  ( y  e.  ( a  i^i  x )  /\  ( (  a  i^i  x )  i^i  y )  =  (/) ) )
28:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  ( [. ( a  i^i  x )  /  b ]. y  e.  b  /\  [. ( a  i^i  x )  /  b ]. ( b  i^i  y )  =  (/) ) )
29:27,28:  |-  ( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
30:29:  |-  A. y ( [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
31:30:  |-  ( E. y [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
32::  |-  ( E. y  e.  ( a