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Theorem List for Metamath Proof Explorer - 27701-27800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreme2ebindVD 27701 The following User's Proof is a Virtual Deduction proof (see: wvd1 27353) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. e2ebind 27345 is e2ebindVD 27701 without virtual deductions and was automatically derived from e2ebindVD 27701
1::  |-  ( ph  <->  ph )
2:1:  |-  ( A. y y  =  x  ->  ( ph  <->  ph ) )
3:2:  |-  ( A. y y  =  x  ->  ( E. y ph  <->  E. x ph  ) )
4::  |-  (. A. y y  =  x  ->.  A. y y  =  x ).
5:3,4:  |-  (. A. y y  =  x  ->.  ( E. y ph  <->  E. x  ph ) ).
6::  |-  ( A. y y  =  x  ->  A. y A. y y  =  x )
7:5,6:  |-  (. A. y y  =  x  ->.  A. y ( E. y ph  <->  E. x ph ) ).
8:7:  |-  (. A. y y  =  x  ->.  ( E. y E. y ph  <->  E. y E. x ph ) ).
9::  |-  ( E. y E. x ph  <->  E. x E. y ph )
10:8,9:  |-  (. A. y y  =  x  ->.  ( E. y E. y ph  <->  E. x E. y ph ) ).
11::  |-  ( E. y ph  ->  A. y E. y ph )
12:11:  |-  ( E. y E. y ph  <->  E. y ph )
13:10,12:  |-  (. A. y y  =  x  ->.  ( E. x E. y ph  <->  E. y ph ) ).
14:13:  |-  ( A. y y  =  x  ->  ( E. x E. y ph  <->  E.  y ph ) )
15::  |-  ( A. y y  =  x  <->  A. x x  =  y )
qed:14,15:  |-  ( A. x x  =  y  ->  ( E. x E. y ph  <->  E.  y ph ) )
(Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( E. x E. y ph  <->  E. y ph )
 )
 
16.22.6  Virtual Deduction transcriptions of textbook proofs
 
Theoremsb5ALTVD 27702* The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Unit 20 Excercise 3.a., which is sb5 1994, found in the "Answers to Starred Exercises" on page 457 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. The same proof may also be interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sb5ALT 27304 is sb5ALTVD 27702 without virtual deductions and was automatically derived from sb5ALTVD 27702.
1::  |-  (. [ y  /  x ] ph  ->.  [ y  /  x ] ph ).
2::  |-  [ y  /  x ] x  =  y
3:1,2:  |-  (. [ y  /  x ] ph  ->.  [ y  /  x ] ( x  =  y  /\  ph ) ).
4:3:  |-  (. [ y  /  x ] ph  ->.  E. x ( x  =  y  /\  ph  ) ).
5:4:  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph )  )
6::  |-  (. E. x ( x  =  y  /\  ph )  ->.  E. x ( x  =  y  /\  ph ) ).
7::  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  ( x  =  y  /\  ph ) ).
8:7:  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  ph ).
9:7:  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  x  =  y ).
10:8,9:  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  [ y  /  x ] ph ).
101::  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
11:101,10:  |-  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph  )
12:5,11:  |-  ( ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph  ) )  /\  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
qed:12:  |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )  )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )
 )
 
Theoremvk15.4jVD 27703 The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Unit 15 Excercise 4.f. found in the "Answers to Starred Exercises" on page 442 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. The same proof may also be interpreted to be a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. vk15.4j 27307 is vk15.4jVD 27703 without virtual deductions and was automatically derived from vk15.4jVD 27703. Step numbers greater than 25 are additional steps necessary for the sequent calculus proof not contained in the Fitch-style proof. Otherwise, step i of the User's Proof corresponds to step i of the Fitch-style proof.
h1::  |-  -.  ( E. x -.  ph  /\  E. x ( ps  /\  -.  ch ) )
h2::  |-  ( A. x ch  ->  -.  E. x ( th  /\  ta  ) )
h3::  |-  -.  A. x ( ta  ->  ph )
4::  |-  (. -.  E. x -.  th  ->.  -.  E. x -.  th ).
5:4:  |-  (. -.  E. x -.  th  ->.  A. x th ).
6:3:  |-  E. x ( ta  /\  -.  ph )
7::  |-  (. -.  E. x -.  th ,. ( ta  /\  -.  ph )  ->.  ( ta  /\  -.  ph ) ).
8:7:  |-  (. -.  E. x -.  th ,. ( ta  /\  -.  ph )  ->.  ta ).
9:7:  |-  (. -.  E. x -.  th ,. ( ta  /\  -.  ph )  ->.  -.  ph ).
10:5:  |-  (. -.  E. x -.  th  ->.  th ).
11:10,8:  |-  (. -.  E. x -.  th ,. ( ta  /\  -.  ph )  ->.  ( th  /\  ta ) ).
12:11:  |-  (. -.  E. x -.  th ,. ( ta  /\  -.  ph )  ->.  E. x ( th  /\  ta ) ).
13:12:  |-  (. -.  E. x -.  th ,. ( ta  /\  -.  ph )  ->.  -.  -.  E. x ( th  /\  ta ) ).
14:2,13:  |-  (. -.  E. x -.  th ,. ( ta  /\  -.  ph )  ->.  -.  A. x ch ).
140::  |-  ( E. x -.  th  ->  A. x E. x -.  th  )
141:140:  |-  ( -.  E. x -.  th  ->  A. x -.  E. x  -.  th )
142::  |-  ( A. x ch  ->  A. x A. x ch )
143:142:  |-  ( -.  A. x ch  ->  A. x -.  A. x ch  )
144:6,14,141,143:  |-  (. -.  E. x -.  th  ->.  -.  A. x ch  ).
15:1:  |-  ( -.  E. x -.  ph  \/  -.  E. x ( ps  /\  -.  ch ) )
16:9:  |-  (. -.  E. x -.  th ,. ( ta  /\  -.  ph )  ->.  E. x -.  ph ).
161::  |-  ( E. x -.  ph  ->  A. x E. x -.  ph  )
162:6,16,141,161:  |-  (. -.  E. x -.  th  ->.  E. x -.  ph  ).
17:162:  |-  (. -.  E. x -.  th  ->.  -.  -.  E. x  -.  ph ).
18:15,17:  |-  (. -.  E. x -.  th  ->.  -.  E. x (  ps  /\  -.  ch ) ).
19:18:  |-  (. -.  E. x -.  th  ->.  A. x ( ps  ->  ch ) ).
20:144:  |-  (. -.  E. x -.  th  ->.  E. x -.  ch  ).
21::  |-  (. -.  E. x -.  th ,. -.  ch  ->.  -.  ch ).
22:19:  |-  (. -.  E. x -.  th  ->.  ( ps  ->  ch  ) ).
23:21,22:  |-  (. -.  E. x -.  th ,. -.  ch  ->.  -.  ps ).
24:23:  |-  (. -.  E. x -.  th ,. -.  ch  ->.  E.  x -.  ps ).
240::  |-  ( E. x -.  ps  ->  A. x E. x -.  ps  )
241:20,24,141,240:  |-  (. -.  E. x -.  th  ->.  E. x -.  ps  ).
25:241:  |-  (. -.  E. x -.  th  ->.  -.  A. x ps  ).
qed:25:  |-  ( -.  E. x -.  th  ->  -.  A. x ps )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  ( E. x  -.  ph  /\ 
 E. x ( ps 
 /\  -.  ch )
 )   &    |-  ( A. x ch  ->  -.  E. x ( th  /\  ta )
 )   &    |- 
 -.  A. x ( ta 
 ->  ph )   =>    |-  ( -.  E. x  -.  th  ->  -.  A. x ps )
 
Theoremnotnot2ALTVD 27704 The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Theorem 5 of Section 14 of [Margaris] p. 59 ( which is notnot2 106). The same proof may also be interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. notnot2ALT 27308 is notnot2ALTVD 27704 without virtual deductions and was automatically derived from notnot2ALTVD 27704. Step i of the User's Proof corresponds to step i of the Fitch-style proof.
1::  |-  (. -.  -.  ph  ->.  -.  -.  ph ).
2::  |-  ( -.  -.  ph  ->  ( -.  ph  ->  -.  -.  -.  ph ) )
3:1:  |-  (. -.  -.  ph  ->.  ( -.  ph  ->  -.  -.  -.  ph ) ).
4::  |-  ( ( -.  ph  ->  -.  -.  -.  ph )  ->  ( -.  -.  ph  ->  ph ) )
5:3:  |-  (. -.  -.  ph  ->.  ( -.  -.  ph  ->  ph ) ).
6:5,1:  |-  (. -.  -.  ph  ->.  ph ).
qed:6:  |-  ( -.  -.  ph  ->  ph )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  -.  ph  ->  ph )
 
Theoremcon3ALTVD 27705 The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Theorem 7 of Section 14 of [Margaris] p. 60 ( which is con3 128). The same proof may also be interpreted to be a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. con3ALT 27309 is con3ALTVD 27705 without virtual deductions and was automatically derived from con3ALTVD 27705. Step i of the User's Proof corresponds to step i of the Fitch-style proof.
1::  |-  (. ( ph  ->  ps )  ->.  ( ph  ->  ps ) ).
2::  |-  (. ( ph  ->  ps ) ,. -.  -.  ph  ->.  -.  -.  ph ).
3::  |-  ( -.  -.  ph  ->  ph )
4:2:  |-  (. ( ph  ->  ps ) ,. -.  -.  ph  ->.  ph ).
5:1,4:  |-  (. ( ph  ->  ps ) ,. -.  -.  ph  ->.  ps ).
6::  |-  ( ps  ->  -.  -.  ps )
7:6,5:  |-  (. ( ph  ->  ps ) ,. -.  -.  ph  ->.  -.  -.  ps ).
8:7:  |-  (. ( ph  ->  ps )  ->.  ( -.  -.  ph  ->  -.  -.  ps  ) ).
9::  |-  ( ( -.  -.  ph  ->  -.  -.  ps )  ->  ( -.  ps  ->  -.  ph ) )
10:8:  |-  (. ( ph  ->  ps )  ->.  ( -.  ps  ->  -.  ph ) ).
qed:10:  |-  ( ( ph  ->  ps )  ->  ( -.  ps  ->  -.  ph ) )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( -.  ps  ->  -.  ph ) )
 
16.22.7  Theorems proved using conjunction-form virtual deduction
 
TheoremelpwgdedVD 27706 Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 3573. In form of VD deduction with  ph and  ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 27346 is elpwgdedVD 27706 using conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (. ph  ->.  A  e.  _V ).   &    |-  (. ps  ->.  A 
 C_  B ).   =>    |-  (. (. ph ,. ps ).  ->.  A  e.  ~P B ).
 
Theoremsspwimp 27707 If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimp 27707, using conventional notation, was translated from virtual deduction form, sspwimpVD 27708, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  B  ->  ~P A  C_ 
 ~P B )
 
TheoremsspwimpVD 27708 The following User's Proof is a Virtual Deduction proof ( see: wvd1 27353) using conjunction-form virtual hypothesis collections. It was completed manually, but has the potential to be completed automatically by a tools program which would invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimp 27707 is sspwimpVD 27708 without virtual deductions and was derived from sspwimpVD 27708. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  (. A  C_  B  ->.  A  C_  B ).
2::  |-  (. ..............  x  e.  ~P A  ->.  x  e.  ~P A ).
3:2:  |-  (. ..............  x  e.  ~P A  ->.  x  C_  A ).
4:3,1:  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  C_  B ).
5::  |-  x  e.  _V
6:4,5:  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  e.  ~P B  ).
7:6:  |-  (. A  C_  B  ->.  ( x  e.  ~P A  ->  x  e.  ~P B )  ).
8:7:  |-  (. A  C_  B  ->.  A. x ( x  e.  ~P A  ->  x  e.  ~P B ) ).
9:8:  |-  (. A  C_  B  ->.  ~P A  C_  ~P B ).
qed:9:  |-  ( A  C_  B  ->  ~P A  C_  ~P B )
 |-  ( A  C_  B  ->  ~P A  C_ 
 ~P B )
 
Theoremsspwimpcf 27709 If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimpcf 27709, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 27710, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  B  ->  ~P A  C_ 
 ~P B )
 
TheoremsspwimpcfVD 27710 The following User's Proof is a Virtual Deduction proof ( see: wvd1 27353) using conjunction-form virtual hypothesis collections. It was completed automatically by a tools program which would invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimpcf 27709 is sspwimpcfVD 27710 without virtual deductions and was derived from sspwimpcfVD 27710. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  (. A  C_  B  ->.  A  C_  B ).
2::  |-  (. ...........  x  e.  ~P A  ->.  x  e.  ~P A ).
3:2:  |-  (. ...........  x  e.  ~P A  ->.  x  C_  A ).
4:3,1:  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  C_  B ).
5::  |-  x  e.  _V
6:4,5:  |-  (. (. A  C_  B ,. x  e.  ~P A ).  ->.  x  e.  ~P B  ).
7:6:  |-  (. A  C_  B  ->.  ( x  e.  ~P A  ->  x  e.  ~P B )  ).
8:7:  |-  (. A  C_  B  ->.  A. x ( x  e.  ~P A  ->  x  e.  ~P B ) ).
9:8:  |-  (. A  C_  B  ->.  ~P A  C_  ~P B ).
qed:9:  |-  ( A  C_  B  ->  ~P A  C_  ~P B )
 |-  ( A  C_  B  ->  ~P A  C_ 
 ~P B )
 
TheoremsuctrALTcf 27711 The sucessor of a transitive class is transitive. suctrALTcf 27711, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 27712, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  suc  A )
 
TheoremsuctrALTcfVD 27712 The following User's Proof is a Virtual Deduction proof ( see: wvd1 27353) using conjunction-form virtual hypothesis collections. The conjunction-form version of completeusersproof.cmd. It allows the User to avoid superflous virtual hypotheses. This proof was completed automatically by a tools program which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. suctrALTcf 27711 is suctrALTcfVD 27712 without virtual deductions and was derived automatically from suctrALTcfVD 27712. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
1::  |-  (. Tr  A  ->.  Tr  A ).
2::  |-  (..........  ( z  e.  y  /\  y  e.  suc  A )  ->.  ( z  e.  y  /\  y  e.  suc  A ) ).
3:2:  |-  (..........  ( z  e.  y  /\  y  e.  suc  A )  ->.  z  e.  y ).
4::  |-  (.................................... .......  y  e.  A  ->.  y  e.  A ).
5:1,3,4:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ,  y  e.  A ).  ->.  z  e.  A ).
6::  |-  A  C_  suc  A
7:5,6:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ,  y  e.  A ).  ->.  z  e.  suc  A ).
8:7:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ).  ->.  ( y  e.  A  ->  z  e.  suc  A ) ).
9::  |-  (.................................... ......  y  =  A  ->.  y  =  A ).
10:3,9:  |-  (.........  (. ( z  e.  y  /\  y  e.  suc  A ) ,  y  =  A ).  ->.  z  e.  A ).
11:10,6:  |-  (.........  (. ( z  e.  y  /\  y  e.  suc  A ) ,  y  =  A ).  ->.  z  e.  suc  A ).
12:11:  |-  (...........  ( z  e.  y  /\  y  e.  suc  A )  ->.  ( y  =  A  ->  z  e.  suc  A ) ).
13:2:  |-  (...........  ( z  e.  y  /\  y  e.  suc  A )  ->.  y  e.  suc  A ).
14:13:  |-  (...........  ( z  e.  y  /\  y  e.  suc  A )  ->.  ( y  e.  A  \/  y  =  A ) ).
15:8,12,14:  |-  (. (. Tr  A ,. ( z  e.  y  /\  y  e.  suc  A )  ).  ->.  z  e.  suc  A ).
16:15:  |-  (. Tr  A  ->.  ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) ).
17:16:  |-  (. Tr  A  ->.  A. z A. y ( ( z  e.  y  /\  y  e.  suc  A )  ->  z  e.  suc  A ) ).
18:17:  |-  (. Tr  A  ->.  Tr  suc  A ).
qed:18:  |-  ( Tr  A  ->  Tr  suc  A )
 |-  ( Tr  A  ->  Tr  suc  A )
 
16.22.8  Theorems with VD proofs in conventional notation derived from VD proofs
 
TheoremsuctrALT3 27713 The successor of a transtive class is transitive. suctrALT3 27713 is the completed proof in conventional notation of the Virtual Deduction proof http://www.virtualdeduction.com/suctralt3vd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 27353 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction ( e.g. , the sub-theorem whose assertion is step 19 used jaoded 27348). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem ( e.g. , the sub-theorem whose assertion is step 24 used dftr2 4055) . (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( Tr  A  ->  Tr  suc  A )
 
TheoremsspwimpALT 27714 If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimpALT 27714 is the completed proof in conventional notation of the Virtual Deduction proof http://www.virtualdeduction.com/sspwimpaltvd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 27353 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction ( e.g. , the sub-theorem whose assertion is step 9 used elpwgded 27346). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem ( e.g. , the sub-theorem whose assertion is step 5 used elpwi 3574) . (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  C_  B  ->  ~P A  C_ 
 ~P B )
 
TheoremunisnALT 27715 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. The User manually input on a mmj2 Proof Worksheet, without labels, all steps of unisnALT 27715 except 1, 11, 15, 21, and 30. With execution of the mmj2 unification command, mmj2 could find labels for all of steps except for 2, 12, 16, 22, and 31 (and the then non-existing steps 1, 11, 15, 21, and 30) . mmj2 could not find reference theorems for those five steps because the hypothesis field of each of these steps was empty and none of those steps unifies with a theorem in set.mm. Each of these five steps is a semantic variation of a theorem in set.mm and is 2-step provable. mmj2 does not have the ability to automatically generate the semantic variation in set.mm of a theorem in an mmj2 Proof Worksheet unless the theorem in the Proof Worksheet is labeled with a 1-hypothesis deduction whose hypothesis is a theorem in set.mm which unifies with the theorem in the Proof Worksheet. The stepprover.c program, which invokes mmj2, has this capability. stepprover.c automatically generated steps 1, 11, 15, 21, and 30, labeled all steps, and generated the RPN proof of unisnALT 27715. Roughly speaking, stepprover.c added to the Proof Worksheet a labeled duplicate step of each non-unifying theorem for each label in a text file, labels.txt, containing a list of labels provided by the User. Upon mmj2 unification, stepprover.c identified a label for each of the five theorems which 2-step proves it. For unisnALT 27715, the label list is a list of all 1-hypothesis propositional calculus deductions in set.mm. stepproverp.c is the same as stepprover.c except that it intermittently pauses during execution, allowing the User to observe the changes to a text file caused by the execution of particular statements of the program. (Contributed by Alan Sare, 19-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |- 
 U. { A }  =  A
 
16.22.9  Theorems with a proof in conventional notation automatically derived

by completeusersproof.c from a Virtual Deduction User's Proof

 
Theoremnotnot2ALT2 27716 Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. (Contributed by Alan Sare, 11-Sep-2016.)
 |-  ( -.  -.  ph  ->  ph )
 
TheoremsuctrALT4 27717 The sucessor of a transitive class is transitive. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in http://www.virtualdeduction.com/suctralt3vd.html. (Contributed by Alan Sare, 11-Sep-2016.)
 |-  ( Tr  A  ->  Tr  suc  A )
 
TheoremsspwimpALT2 27718 If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in http://www.virtualdeduction.com/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.)
 |-  ( A  C_  B  ->  ~P A  C_ 
 ~P B )
 
Theoreme2ebindALT 27719 Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in e2ebindVD 27701. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( E. x E. y ph  <->  E. y ph )
 )
 
Theorema9e2ndALT 27720* If at least two sets exist (dtru 4139) , then the same is true expressed in an alternate form similar to the form of a9e 1817. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in a9e2ndVD 27697. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
 
Theorema9e2ndeqALT 27721* "At least two sets exist" expressed in the form of dtru 4139 is logically equivalent to the same expressed in a form similar to a9e 1817 if dtru 4139 is false implies  u  =  v. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in a9e2ndeqVD 27698. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  A. x  x  =  y  \/  u  =  v )  <->  E. x E. y ( x  =  u  /\  y  =  v )
 )
 
Theorem2sb5ndALT 27722* Equivalence for double substitution 2sb5 2075 without distinct  x,  y requirement. 2sb5nd 27342 is derived from 2sb5ndVD 27699. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in 2sb5ndVD 27699. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( -.  A. x  x  =  y  \/  u  =  v )  ->  ( [ u  /  x ] [ v  /  y ] ph  <->  E. x E. y
 ( ( x  =  u  /\  y  =  v )  /\  ph )
 ) )
 
16.23  Mathbox for Jonathan Ben-Naim

Note: On 4-Sep-2016 and after, 745 unused theorems were deleted from this mathbox, and 359 theorems used only once or twice were merged into their referencing theorems. The originals can be recovered from set.mm versions prior to this date.

 
Syntaxw-bnj17 27723 Extend wff notation with the 4-way conjunction. (New usage is discouraged.)
 wff  ( ph  /\  ps  /\  ch 
 /\  th )
 
Definitiondf-bnj17 27724 Define the 4-way conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ps  /\  ch )  /\  th ) )
 
Syntaxc-bnj14 27725 Extend class notation with the function giving: the class of all elements of  A that are "smaller" than  X according to  R. (New usage is discouraged.)
 class  pred ( X ,  A ,  R )
 
Definitiondf-bnj14 27726* Define the function giving: the class of all elements of  A that are "smaller" than  X according to  R. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  pred ( X ,  A ,  R )  =  {
 y  e.  A  |  y R X }
 
Syntaxw-bnj13 27727 Extend wff notation with the following predicate:  R is set-like on  A. (New usage is discouraged.)
 wff  R  Se  A
 
Definitiondf-bnj13 27728* Define the following predicate:  R is set-like on  A. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( R  Se  A  <->  A. x  e.  A  pred ( x ,  A ,  R )  e.  _V )
 
Syntaxw-bnj15 27729 Extend wff notation with the following predicate:  R is both well-founded and set-like on 
A. (New usage is discouraged.)
 wff  R 
 FrSe  A
 
Definitiondf-bnj15 27730 Define the following predicate:  R is both well-founded and set-like on  A. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( R  FrSe  A  <->  ( R  Fr  A  /\  R  Se  A ) )
 
Syntaxc-bnj18 27731 Extend class notation with the function giving: the transitive closure of  X in  A by  R. (New usage is discouraged.)
 class  trCl ( X ,  A ,  R )
 
Definitiondf-bnj18 27732* Define the function giving: the transitive closure of  X in  A by  R. This definition has been designed for facilitating verification that it is eliminable and that the $d restrictions are sound and complete. For a more readable definition see bnj882 27970. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  trCl ( X ,  A ,  R )  =  U_ f  e.  { f  |  E. n  e.  ( om  \  { (/) } ) ( f  Fn  n  /\  ( f `  (/) )  = 
 pred ( X ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) ) } U_ i  e.  dom  f ( f `  i )
 
Syntaxw-bnj19 27733 Extend wff notation with the following predicate:  B is transitive for  A and  R. (New usage is discouraged.)
 wff  TrFo
 ( B ,  A ,  R )
 
Definitiondf-bnj19 27734* Define the following predicate:  B is transitive for  A and  R. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (  TrFo ( B ,  A ,  R )  <->  A. x  e.  B  pred ( x ,  A ,  R )  C_  B )
 
16.23.1  First order logic and set theory
 
Theorembnj170 27735  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  <->  ( ( ps 
 /\  ch )  /\  ph )
 )
 
Theorembnj240 27736  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps  ->  ps' )   &    |-  ( ch  ->  ch' )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( ps'  /\  ch' ) )
 
Theorembnj248 27737  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( (
 ph  /\  ps )  /\  ch )  /\  th ) )
 
Theorembnj250 27738  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  (
 ( ps  /\  ch )  /\  th ) ) )
 
Theorembnj251 27739  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ( ps  /\  ( ch  /\  th ) ) ) )
 
Theorembnj252 27740  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ( ps  /\  ch  /\  th ) ) )
 
Theorembnj253 27741  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th ) )
 
Theorembnj255 27742  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ps  /\  ( ch  /\  th ) ) )
 
Theorembnj256 27743  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) )
 
Theorembnj257 27744  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ps  /\ 
 th  /\  ch )
 )
 
Theorembnj258 27745  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ps  /\  th )  /\  ch ) )
 
Theorembnj268 27746  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ch  /\ 
 ps  /\  th )
 )
 
Theorembnj290 27747  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ph  /\  ch  /\ 
 th  /\  ps )
 )
 
Theorembnj291 27748  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ph  /\ 
 ch  /\  th )  /\  ps ) )
 
Theorembnj312 27749  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ps  /\  ph 
 /\  ch  /\  th )
 )
 
Theorembnj334 27750  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ch  /\  ph 
 /\  ps  /\  th )
 )
 
Theorembnj345 27751  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( th  /\  ph 
 /\  ps  /\  ch )
 )
 
Theorembnj422 27752  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ch  /\  th 
 /\  ph  /\  ps )
 )
 
Theorembnj432 27753  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ch 
 /\  th )  /\  ( ph  /\  ps ) ) )
 
Theorembnj446 27754  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  <->  ( ( ps 
 /\  ch  /\  th )  /\  ph ) )
 
Theorembnj21 27755* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { x  e.  A  |  ph }   =>    |-  B  C_  A
 
Theorembnj23 27756* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  B  =  { x  e.  A  |  -.  ph }   =>    |-  ( A. z  e.  B  -.  z R y  ->  A. w  e.  A  ( w R y  ->  [. w  /  x ]. ph ) )
 
Theorembnj31 27757 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x  e.  A  ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  E. x  e.  A  ch )
 
Theorembnj62 27758* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( [. z  /  x ]. x  Fn  A  <->  z  Fn  A )
 
Theorembnj89 27759* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  Z  e.  _V   =>    |-  ( [. Z  /  y ]. E! x ph  <->  E! x [. Z  /  y ]. ph )
 
Theorembnj90 27760* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  Y  e.  _V   =>    |-  ( [. Y  /  x ]. z  Fn  x  <->  z  Fn  Y )
 
Theorembnj101 27761 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  E. x ph   &    |-  ( ph  ->  ps )   =>    |-  E. x ps
 
Theorembnj105 27762 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  1o  e.  _V
 
Theorembnj115 27763 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <-> 
 A. n  e.  D  ( ta  ->  th )
 )   =>    |-  ( et  <->  A. n ( ( n  e.  D  /\  ta )  ->  th )
 )
 
Theorembnj132 27764* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 E. x ( ps 
 ->  ch ) )   =>    |-  ( ph  <->  ( ps  ->  E. x ch ) )
 
Theorembnj133 27765 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 E. x ps )   &    |-  ( ch 
 <->  ps )   =>    |-  ( ph  <->  E. x ch )
 
Theorembnj142 27766 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( F  Fn  { A }  ->  ( u  e.  F  ->  u  =  <. A ,  ( F `  A )
 >. ) )
 
Theorembnj145 27767 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  ( F `  A )  e.  _V   =>    |-  ( F  Fn  { A }  ->  F  =  { <. A ,  ( F `  A ) >. } )
 
Theorembnj156 27768 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ze0  <->  ( f  Fn  1o  /\  ph'  /\  ps' ) )   &    |-  ( ze1  <->  [. g  /  f ]. ze0 )   &    |-  ( ph1  <->  [. g  /  f ]. ph' )   &    |-  ( ps1  <->  [. g  /  f ]. ps' )   =>    |-  ( ze1  <->  ( g  Fn 
 1o  /\  ph1  /\  ps1 ) )
 
Theorembnj158 27769* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( m  e.  D  ->  E. p  e.  om  m  =  suc  p )
 
Theorembnj168 27770* First-order logic and set theory. Revised to remove dependence on ax-reg 7239. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( ( n  =/= 
 1o  /\  n  e.  D )  ->  E. m  e.  D  n  =  suc  m )
 
Theorembnj206 27771 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph'  <->  [. M  /  n ]. ph )   &    |-  ( ps'  <->  [. M  /  n ].
 ps )   &    |-  ( ch'  <->  [. M  /  n ].
 ch )   &    |-  M  e.  _V   =>    |-  ( [. M  /  n ]. ( ph  /\  ps  /\ 
 ch )  <->  ( ph'  /\  ps'  /\  ch' ) )
 
Theorembnj216 27772 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  e.  _V   =>    |-  ( A  =  suc  B 
 ->  B  e.  A )
 
Theorembnj219 27773 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( n  =  suc  m  ->  m  _E  n )
 
Theorembnj226 27774* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  C_  C   =>    |-  U_ x  e.  A  B  C_  C
 
Theorembnj228 27775 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 A. x  e.  A  ps )   =>    |-  ( ( x  e.  A  /\  ph )  ->  ps )
 
Theorembnj519 27776 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( B  e.  _V  ->  Fun  { <. A ,  B >. } )
 
Theorembnj521 27777 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( A  i^i  { A }
 )  =  (/)
 
Theorembnj524 27778 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ps )   &    |-  A  e.  _V   =>    |-  ( [. A  /  x ].
 ph 
 <-> 
 [. A  /  x ].
 ps )
 
Theorembnj525 27779* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. ph  <->  ph )
 
Theorembnj534 27780* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  ( E. x ph 
 /\  ps ) )   =>    |-  ( ch  ->  E. x ( ph  /\  ps ) )
 
Theorembnj538 27781* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( [. A  /  y ]. A. x  e.  B  ph  <->  A. x  e.  B  [. A  /  y ]. ph )
 
Theorembnj529 27782 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( M  e.  D  -> 
 (/)  e.  M )
 
Theorembnj551 27783 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( m  =  suc  p 
 /\  m  =  suc  i )  ->  p  =  i )
 
Theorembnj563 27784 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   =>    |-  ( ( et  /\  rh )  ->  suc  i  e.  m )
 
Theorembnj564 27785 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ta 
 <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   =>    |-  ( ta  ->  dom  f  =  m )
 
Theorembnj593 27786 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  E. x ps )   &    |-  ( ps  ->  ch )   =>    |-  ( ph  ->  E. x ch )
 
Theorembnj596 27787 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  E. x ps )   =>    |-  ( ph  ->  E. x ( ph  /\ 
 ps ) )
 
Theorembnj610 27788* Pass from equality ( x  =  A) to substitution ( [. A  /  x ].) without the distinct variable restriction ($d  A  x). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  ( ph  <->  ps' ) )   &    |-  ( y  =  A  ->  ( ps'  <->  ps ) )   =>    |-  ( [. A  /  x ]. ph  <->  ps )
 
Theorembnj642 27789  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ph )
 
Theorembnj643 27790  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ps )
 
Theorembnj645 27791  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  th )
 
Theorembnj658 27792  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ( ph  /\  ps  /\  ch ) )
 
Theorembnj667 27793  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ( ps  /\  ch  /\  th ) )
 
Theorembnj705 27794  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj706 27795  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj707 27796  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj708 27797  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 ch  /\  th )  ->  ta )
 
Theorembnj721 27798  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ta )   =>    |-  (
 ( ph  /\  ps  /\  ch 
 /\  th )  ->  ta )
 
Theorembnj832 27799  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps ) )   &    |-  ( ph  ->  ta )   =>    |-  ( et  ->  ta )
 
Theorembnj833 27800  /\-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ph  /\  ps ) )   &    |-  ( ps  ->  ta )   =>    |-  ( et  ->  ta )
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