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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelex2VD 27401* Virtual deduction proof of elex2 2739. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  E. x  x  e.  B )
 
Theoremelex22VD 27402* Virtual deduction proof of elex22 2738. (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 27403* Virtual deduction proof of eqsbc3r 2978. (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 27404* Virtual deduction proof of zfregs2 7299. (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 27405 Virtual deduction proof of tpid3g 3645. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  B  ->  A  e.  { C ,  D ,  A }
 )
 
Theoremen3lplem1VD 27406* Virtual deduction proof of en3lplem1 7300. (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 27407* Virtual deduction proof of en3lplem2 7301. (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 27408 Virtual deduction proof of en3lp 7302. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  ( A  e.  B  /\  B  e.  C  /\  C  e.  A )
 
16.21.5  Theorems proved using virtual deduction with mmj2 assistance
 
Theoremsimplbi2VD 27409 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_ 27334  |-  ( ( ps  /\  ch )  ->  ph )
qed:3,?: e0_ 27334  |-  ( 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 27410 Virtual deduction proof of 3ornot23 27060. 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_ 27186  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ph ).
4:1,?: e1_ 27186  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ps ).
5:3,4,?: e11 27247  |-  (. ( -.  ph  /\  -.  ps )  ->.  -.  ( ph  \/  ps ) ).
6:2,?: e2 27190  |-  (. ( -.  ph  /\  -.  ps ) ,. ( ch  \/  ph  \/  ps )  ->.  ( ch  \/  ( ph  \/  ps ) ) ).
7:5,6,?: e12 27286  |-  (. ( -.  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 27411 Virtual deduction proof of orbi1r 27061. 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 27190  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ph  \/  ch ) ).
4:1,3,?: e12 27286  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ps  \/  ch ) ).
5:4,?: e2 27190  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ph )  ->.  ( ch  \/  ps ) ).
6:5:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ph )  ->  ( ch  \/  ps ) ) ).
7::  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ps ) ).
8:7,?: e2 27190  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ps  \/  ch ) ).
9:1,8,?: e12 27286  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ph  \/  ch ) ).
10:9,?: e2 27190  |-  (. ( ph  <->  ps ) ,. ( ch  \/  ps )  ->.  ( ch  \/  ph ) ).
11:10:  |-  (. ( ph  <->  ps )  ->.  ( ( ch  \/  ps )  ->  ( ch  \/  ph ) ) ).
12:6,11,?: e11 27247  |-  (. ( 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 27412 Virtual deduction proof of bitr3 27062. 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_ 27186  |-  (. ( ph  <->  ps )  ->.  ( ps  <->  ph ) ).
3::  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ph  <->  ch ) ).
4:3,?: e2 27190  |-  (. ( ph  <->  ps ) ,. ( ph  <->  ch )  ->.  ( ch  <->  ph ) ).
5:2,4,?: e12 27286  |-  (. ( 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 27413 Virtual deduction proof of 3orbi123 27063. 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_ 27186  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ph  <->  ps ) ).
3:1,?: e1_ 27186  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ch  <->  th ) ).
4:1,?: e1_ 27186  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ta  <->  et ) ).
5:2,3,?: e11 27247  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch )  <->  ( ps  \/  th ) ) ).
6:5,4,?: e11 27247  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
7:?:  |-  ( ( ( ph  \/  ch )  \/  ta )  <->  ( ph  \/  ch  \/  ta ) )
8:6,7,?: e10 27254  |-  (. ( ( ph  <->  ps )  /\  ( ch  <->  th )  /\  ( ta  <->  et ) )  ->.  ( ( ph  \/  ch  \/  ta )  <->  ( ( ps  \/  th )  \/  et ) ) ).
9:?:  |-  ( ( ( ps  \/  th )  \/  et )  <->  ( ps  \/  th  \/  et ) )
10:8,9,?: e10 27254  |-  (. ( ( 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 27414 Virtual deduction proof of sbc3org 27085. 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_ 27186  |-  (. 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 27254  |-  (. A  e.  B  ->.  [. A  /  x ]. ( ( ( ph  \/  ps )  \/  ch )  <->  ( ph  \/  ps  \/  ch ) ) ).
4:1,33,?: e11 27247  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ( ph  \/  ps )  \/  ch )  <->  [. A  /  x ]. ( ph  \/  ps  \/  ch ) ) ).
5:2,4,?: e11 27247  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps  \/  ch )  <->  ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch ) ) ).
6:1,?: e1_ 27186  |-  (. A  e.  B  ->.  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) ).
7:6,?: e1_ 27186  |-  (. A  e.  B  ->.  ( ( [. A  /  x ]. ( ph  \/  ps )  \/  [. A  /  x ]. ch )  <->  ( ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps )  \/  [. A  /  x ]. ch ) ) ).
8:5,7,?: e11 27247  |-  (. 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 27254  |-  (. 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 27415* Virtual deduction proof of alrim3con13v 27086. 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 27190  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ps ).
4:2,?: e2 27190  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ph ).
5:2,?: e2 27190  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ch ).
6:1,4,?: e12 27286  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ph ).
7:3,?: e2 27190  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ps ).
8:5,?: e2 27190  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  A. x ch ).
9:7,6,8,?: e222 27195  |-  (. ( ph  ->  A. x ph ) ,. ( ps  /\  ph  /\  ch )  ->.  ( A. x ps  /\  A. x ph  /\  A. x ch ) ).
10:9,?: e2 27190  |-  (. ( 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 27416 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 27286  |-  (. ( ( ph  /\  ps )  ->  ( ch  <->  th ) ) ,  ( ph  /\  ps )  ->.  ( ch  <->  th ) ).
6:3,5,?: e32 27320  |-  (. ( ( 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_ 27186  |-  (. ( ( 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 27417 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 27254  |-  (. ph  ->.  ( ps  ->  ( ch  <->  th ) ) ).
6:3,5,?: e21 27292  |-  (. ph ,. ps  ->.  ( ch  <->  th ) ).
7:4,6,?: e32 27320  |-  (. 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 27418* Virtual deduction proof of ra4sbc2 27087. 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 27310  |-  (. 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 27310  |-  (. 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 27317  |-  (. 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 27419 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 27254  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
4:3,?: e1_ 27186  |-  (. ( ( ph  /\  ps  /\  ch )  ->  th )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
5:4,?: e1_ 27186  |-  (. ( ( 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_ 27186  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps )  ->  ( ch  ->  th ) ) ).
9:8,?: e1_ 27186  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ( ph  /\  ps )  /\  ch )  ->  th ) ).
10:2,9,?: e01 27250  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  th ) ).
11:10:  |-  ( ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )  ->  ( ( ph  /\  ps  /\  ch )  ->  th ) )
qed:6,11,?: e00 27330  |-  ( ( ( 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 27420 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 27254  |-  (. ( ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
4:3,?: e1_ 27186  |-  (. ( ( 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_ 27186  |-  (. ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )  ->.  ( ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) ) ).
8:7,2,?: e10 27254  |-  (. ( 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 27330  |-  ( ( ( 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 27421 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_ 27334  |-  ( 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 27422* Virtual deduction proof of sbcoreleleq 27088. 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_ 27186  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  e.  y  <->  x  e.  A ) ).
3:1,?: e1_ 27186  |-  (. A  e.  B  ->.  ( [. A  /  y ]. y  e.  x  <->  A  e.  x ) ).
4:1,?: e1_ 27186  |-  (. A  e.  B  ->.  ( [. A  /  y ]. x  =  y  <->  x  =  A ) ).
5:2,3,4,?: e111 27233  |-  (. 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_ 27186  |-  (. 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 27247  |-  (. 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 27423* Virtual deduction proof of nfra2 2559. 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 27330  |-  ( 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_ 27334  |-  ( 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 27330  |-  ( 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 27424* Virtual deduction proof of tratrb 27089. 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_ 27186  |-  (. ( 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_ 27186  |-  (. ( 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_ 27186  |-  (. ( 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 27190  |-  (. ( 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 27190  |-  (. ( 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 27215  |-  (. ( 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 27212  |-  (. ( 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 27194  |-  (. ( 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 27289  |-  (. ( 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 27317  |-  (. ( 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 27317  |-  (. ( 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 27289  |-  (. ( 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_ 27186  |-  (. ( 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 27215  |-  (. ( 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 27190  |-  (. ( 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 27286  |-  (. ( 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 27195  |-  (. ( 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_ 27334  |-  ( ( 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_ 27334  |-  ( ( 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_ 27186  |-  (. ( 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 27425 Virtual deduction proof of 3ax5 27090. 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_ 27186  |-  (. 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 27254  |-  (. 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 27426 Virtual deduction proof of syl5imp 27064. 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_ 27186  |-  (. ( ph  ->  ( ps  ->  ch ) )  ->.  ( ps  ->  ( ph  ->  ch ) ) ).
3::  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( th  ->  ps ) ).
4:3,2,?: e21 27292  |-  (. ( ph  ->  ( ps  ->  ch ) ) ,. ( th  ->  ps )  ->.  ( th  ->  ( ph  ->  ch ) ) ).
5:4,?: e2 27190  |-  (. ( 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 27427 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_ 27334  |-  ph
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   =>    |-  ph
 
TheoremancomsimpVD 27428 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_ 27334  |-  ( ( ( 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 27429* Quantification restricted to a subclass for two quantifiers. ssralv 3158 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 27084 is ssralv2VD 27429 without virtual deductions and was automatically derived from ssralv2VD 27429.
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 27430 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4307 using the Axiom of Regularity indirectly through dford2 7205. 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 27091 is ordelordALTVD 27430 without virtual deductions and was automatically derived from ordelordALTVD 27430 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 27431 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 3230 is equncomVD 27431 without virtual deductions and was automatically derived from equncomVD 27431.
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 27432 Inference form of equncom 3230. 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 3231 is equncomiVD 27432 without virtual deductions and was automatically derived from equncomiVD 27432.
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 27433 A set belongs to its successor. Alternate proof of sucid 4364. 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 27434 is sucidALTVD 27433 without virtual deductions and was automatically derived from sucidALTVD 27433. 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 4291, 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 7205.
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 27434 A set belongs to its successor. This proof was automatically derived from sucidALTVD 27433 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 27435 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 4364 is sucidVD 27435 without virtual deductions and was automatically derived from sucidVD 27435.
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 27436 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 27072 is imbi12VD 27436 without virtual deductions and was automatically derived from imbi12VD 27436.
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 27437 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 27073 is imbi13VD 27437 without virtual deductions and was automatically derived from imbi13VD 27437.
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 27438 Distribution of class substitution over a left-nested implication. Similar to sbcimg 2962. 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 27092 is sbcim2gVD 27438 without virtual deductions and was automatically derived from sbcim2gVD 27438.
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 27439 Implication form of sbcbiiOLD 2977. 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 27093 is sbcbiVD 27439 without virtual deductions and was automatically derived from sbcbiVD 27439.
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 27440* 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 27094 is trsbcVD 27440 without virtual deductions and was automatically derived from trsbcVD 27440.
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 27441* 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 27095 is truniALTVD 27441 without virtual deductions and was automatically derived from truniALTVD 27441.
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 27442 Non-virtual deduction form of e33 27296. 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 27074 is ee33VD 27442 without virtual deductions and was automatically derived from ee33VD 27442.
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 27443* The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 27444. 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 27444 is trintALTVD 27443 without virtual deductions and was automatically derived from trintALTVD 27443.
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 27444* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 27444 is an alternative proof of trint 4025. trintALT 27444 is trintALTVD 27443 without virtual deductions and was automatically derived from trintALTVD 27443 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 27445 The first equality of Exercise 13 of [TakeutiZaring] p. 22. Virtual deduction proof of undif3 3336. 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 3336 is undif3VD 27445 without virtual deductions and was automatically derived from undif3VD 27445.
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 27446 Virtual deduction proof of sbcss 27096. 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 27096 is sbcssVD 27446 without virtual deductions and was automatically derived from sbcssVD 27446.
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 27447 Virtual deduction proof of csbing 3283. 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 3283 is csbingVD 27447 without virtual deductions and was automatically derived from csbingVD 27447.
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 27448* Virtual deduction proof of onfrALTlem5 27097. 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 27097 is onfrALTlem5VD 27448 without virtual deductions and was automatically derived from onfrALTlem5VD 27448.
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  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/)  <->  E. y ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/)  ) )
33:31,32:  |-  ( E. y [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) )
34:2:  |-  ( E. y [. ( a  i^i  x )  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  (  b  i^i  y )  =  (/) ) )
35:33,34:  |-  ( [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y  )  =  (/) )
36::  |-  ( E. y  e.  b ( b  i^i  y )  =  (/)  <->  E. y  ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
37:36:  |-  A. b ( E. y  e.  b ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
38:2,37:  |-  ( [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/)  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
39:35,38:  |-  ( [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/)  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) )
40:16,39:  |-  ( ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
41:2:  |-  ( [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( [. ( a  i^i  x )  /  b ]. ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  /  b ]. E. y  e.  b ( b  i^i  y )  =  (/) ) )
qed:40,41:  |-  ( [. ( a  i^i  x )  /  b ]. ( ( b  C_  ( a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x )  C_  ( a  i^i  x )  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x  ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [. ( a  i^i  x )  /  b ]. (
 ( b  C_  (
 a  i^i  x )  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) )  <->  ( ( ( a  i^i  x ) 
 C_  ( a  i^i 
 x )  /\  (
 a  i^i  x )  =/= 
 (/) )  ->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
 
TheoremonfrALTlem4VD 27449* Virtual deduction proof of onfrALTlem4 27098. 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. onfrALTlem4 27098 is onfrALTlem4VD 27449 without virtual deductions and was automatically derived from onfrALTlem4VD 27449.
1::  |-  y  e.  _V
2:1:  |-  ( [. y  /  x ]. ( a  i^i  x )  =  (/)  <->  [_  y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_ (/) )
3:1:  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_  a  i^i  [_ y  /  x ]_ x )
4:1:  |-  [_ y  /  x ]_ a  =  a
5:1:  |-  [_ y  /  x ]_ x  =  y
6:4,5:  |-  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )  =  (  a  i^i  y )
7:3,6:  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( a  i^i  y )
8:1:  |-  [_ y  /  x ]_ (/)  =  (/)
9:7,8:  |-  ( [_ y  /  x ]_ ( a  i^i  x )