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Theorem ax12eq 32576
Description: Basis step for constructing a substitution instance of ax-c15 32525 without using ax-c15 32525. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax12eq  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )

Proof of Theorem ax12eq
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1740 . . 3  |-  ( A. x ( x  =  z  /\  x  =  w )  <->  ( A. x  x  =  z  /\  A. x  x  =  w ) )
2 equid 1863 . . . . . . . 8  |-  x  =  x
32a1i 11 . . . . . . 7  |-  ( x  =  y  ->  x  =  x )
43ax-gen 1677 . . . . . 6  |-  A. x
( x  =  y  ->  x  =  x )
54a1i 11 . . . . 5  |-  ( x  =  x  ->  A. x
( x  =  y  ->  x  =  x ) )
6 equequ1 1875 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  =  x  <->  z  =  x ) )
7 equequ2 1876 . . . . . . . . 9  |-  ( x  =  w  ->  (
z  =  x  <->  z  =  w ) )
86, 7sylan9bb 714 . . . . . . . 8  |-  ( ( x  =  z  /\  x  =  w )  ->  ( x  =  x  <-> 
z  =  w ) )
98sps-o 32543 . . . . . . 7  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  (
x  =  x  <->  z  =  w ) )
10 nfa1-o 32550 . . . . . . . 8  |-  F/ x A. x ( x  =  z  /\  x  =  w )
119imbi2d 323 . . . . . . . 8  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  (
( x  =  y  ->  x  =  x )  <->  ( x  =  y  ->  z  =  w ) ) )
1210, 11albid 1983 . . . . . . 7  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  ( A. x ( x  =  y  ->  x  =  x )  <->  A. x
( x  =  y  ->  z  =  w ) ) )
139, 12imbi12d 327 . . . . . 6  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  (
( x  =  x  ->  A. x ( x  =  y  ->  x  =  x ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
1413adantr 472 . . . . 5  |-  ( ( A. x ( x  =  z  /\  x  =  w )  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( ( x  =  x  ->  A. x
( x  =  y  ->  x  =  x ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
155, 14mpbii 216 . . . 4  |-  ( ( A. x ( x  =  z  /\  x  =  w )  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) )
1615exp32 616 . . 3  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) ) )
171, 16sylbir 218 . 2  |-  ( ( A. x  x  =  z  /\  A. x  x  =  w )  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) ) ) )
18 equequ1 1875 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  w  <->  y  =  w ) )
1918ad2antll 743 . . . . . 6  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( x  =  w  <-> 
y  =  w ) )
20 axc9 2154 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  w  ->  ( y  =  w  ->  A. x  y  =  w )
) )
2120impcom 437 . . . . . . . 8  |-  ( ( -.  A. x  x  =  w  /\  -.  A. x  x  =  y )  ->  ( y  =  w  ->  A. x  y  =  w )
)
2221adantrr 731 . . . . . . 7  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( y  =  w  ->  A. x  y  =  w ) )
23 equtrr 1874 . . . . . . . 8  |-  ( y  =  w  ->  (
x  =  y  ->  x  =  w )
)
2423alimi 1692 . . . . . . 7  |-  ( A. x  y  =  w  ->  A. x ( x  =  y  ->  x  =  w ) )
2522, 24syl6 33 . . . . . 6  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( y  =  w  ->  A. x ( x  =  y  ->  x  =  w ) ) )
2619, 25sylbid 223 . . . . 5  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( x  =  w  ->  A. x ( x  =  y  ->  x  =  w ) ) )
2726adantll 728 . . . 4  |-  ( ( ( A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( x  =  w  ->  A. x ( x  =  y  ->  x  =  w ) ) )
28 equequ1 1875 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  w  <->  z  =  w ) )
2928sps-o 32543 . . . . . 6  |-  ( A. x  x  =  z  ->  ( x  =  w  <-> 
z  =  w ) )
3029imbi2d 323 . . . . . . 7  |-  ( A. x  x  =  z  ->  ( ( x  =  y  ->  x  =  w )  <->  ( x  =  y  ->  z  =  w ) ) )
3130dral2-o 32565 . . . . . 6  |-  ( A. x  x  =  z  ->  ( A. x ( x  =  y  ->  x  =  w )  <->  A. x ( x  =  y  ->  z  =  w ) ) )
3229, 31imbi12d 327 . . . . 5  |-  ( A. x  x  =  z  ->  ( ( x  =  w  ->  A. x
( x  =  y  ->  x  =  w ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
3332ad2antrr 740 . . . 4  |-  ( ( ( A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( ( x  =  w  ->  A. x
( x  =  y  ->  x  =  w ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
3427, 33mpbid 215 . . 3  |-  ( ( ( A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) )
3534exp32 616 . 2  |-  ( ( A. x  x  =  z  /\  -.  A. x  x  =  w
)  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) ) )
36 equequ2 1876 . . . . . . 7  |-  ( x  =  y  ->  (
z  =  x  <->  z  =  y ) )
3736ad2antll 743 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  x  <-> 
z  =  y ) )
38 axc9 2154 . . . . . . . . 9  |-  ( -. 
A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
) )
3938imp 436 . . . . . . . 8  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( z  =  y  ->  A. x  z  =  y )
)
4039adantrr 731 . . . . . . 7  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  y  ->  A. x  z  =  y ) )
4136biimprcd 233 . . . . . . . 8  |-  ( z  =  y  ->  (
x  =  y  -> 
z  =  x ) )
4241alimi 1692 . . . . . . 7  |-  ( A. x  z  =  y  ->  A. x ( x  =  y  ->  z  =  x ) )
4340, 42syl6 33 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  y  ->  A. x ( x  =  y  ->  z  =  x ) ) )
4437, 43sylbid 223 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  x  ->  A. x ( x  =  y  ->  z  =  x ) ) )
4544adantlr 729 . . . 4  |-  ( ( ( -.  A. x  x  =  z  /\  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  x  ->  A. x ( x  =  y  ->  z  =  x ) ) )
467sps-o 32543 . . . . . 6  |-  ( A. x  x  =  w  ->  ( z  =  x  <-> 
z  =  w ) )
4746imbi2d 323 . . . . . . 7  |-  ( A. x  x  =  w  ->  ( ( x  =  y  ->  z  =  x )  <->  ( x  =  y  ->  z  =  w ) ) )
4847dral2-o 32565 . . . . . 6  |-  ( A. x  x  =  w  ->  ( A. x ( x  =  y  -> 
z  =  x )  <->  A. x ( x  =  y  ->  z  =  w ) ) )
4946, 48imbi12d 327 . . . . 5  |-  ( A. x  x  =  w  ->  ( ( z  =  x  ->  A. x
( x  =  y  ->  z  =  x ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
5049ad2antlr 741 . . . 4  |-  ( ( ( -.  A. x  x  =  z  /\  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( ( z  =  x  ->  A. x
( x  =  y  ->  z  =  x ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
5145, 50mpbid 215 . . 3  |-  ( ( ( -.  A. x  x  =  z  /\  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) )
5251exp32 616 . 2  |-  ( ( -.  A. x  x  =  z  /\  A. x  x  =  w
)  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) ) )
53 ax6ev 1815 . . . . 5  |-  E. u  u  =  w
54 ax6ev 1815 . . . . . . 7  |-  E. v 
v  =  z
55 ax-1 6 . . . . . . . . . . 11  |-  ( v  =  u  ->  (
x  =  y  -> 
v  =  u ) )
5655alrimiv 1781 . . . . . . . . . 10  |-  ( v  =  u  ->  A. x
( x  =  y  ->  v  =  u ) )
57 equequ1 1875 . . . . . . . . . . . . 13  |-  ( v  =  z  ->  (
v  =  u  <->  z  =  u ) )
58 equequ2 1876 . . . . . . . . . . . . 13  |-  ( u  =  w  ->  (
z  =  u  <->  z  =  w ) )
5957, 58sylan9bb 714 . . . . . . . . . . . 12  |-  ( ( v  =  z  /\  u  =  w )  ->  ( v  =  u  <-> 
z  =  w ) )
6059adantl 473 . . . . . . . . . . 11  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( v  =  u  <->  z  =  w ) )
61 dveeq2-o 32568 . . . . . . . . . . . . . . 15  |-  ( -. 
A. x  x  =  z  ->  ( v  =  z  ->  A. x  v  =  z )
)
62 dveeq2-o 32568 . . . . . . . . . . . . . . 15  |-  ( -. 
A. x  x  =  w  ->  ( u  =  w  ->  A. x  u  =  w )
)
6361, 62im2anan9 853 . . . . . . . . . . . . . 14  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( (
v  =  z  /\  u  =  w )  ->  ( A. x  v  =  z  /\  A. x  u  =  w
) ) )
6463imp 436 . . . . . . . . . . . . 13  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( A. x  v  =  z  /\  A. x  u  =  w ) )
65 19.26 1740 . . . . . . . . . . . . 13  |-  ( A. x ( v  =  z  /\  u  =  w )  <->  ( A. x  v  =  z  /\  A. x  u  =  w ) )
6664, 65sylibr 217 . . . . . . . . . . . 12  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  A. x
( v  =  z  /\  u  =  w ) )
67 nfa1-o 32550 . . . . . . . . . . . . 13  |-  F/ x A. x ( v  =  z  /\  u  =  w )
6859sps-o 32543 . . . . . . . . . . . . . 14  |-  ( A. x ( v  =  z  /\  u  =  w )  ->  (
v  =  u  <->  z  =  w ) )
6968imbi2d 323 . . . . . . . . . . . . 13  |-  ( A. x ( v  =  z  /\  u  =  w )  ->  (
( x  =  y  ->  v  =  u )  <->  ( x  =  y  ->  z  =  w ) ) )
7067, 69albid 1983 . . . . . . . . . . . 12  |-  ( A. x ( v  =  z  /\  u  =  w )  ->  ( A. x ( x  =  y  ->  v  =  u )  <->  A. x
( x  =  y  ->  z  =  w ) ) )
7166, 70syl 17 . . . . . . . . . . 11  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( A. x ( x  =  y  ->  v  =  u )  <->  A. x
( x  =  y  ->  z  =  w ) ) )
7260, 71imbi12d 327 . . . . . . . . . 10  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( (
v  =  u  ->  A. x ( x  =  y  ->  v  =  u ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
7356, 72mpbii 216 . . . . . . . . 9  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) )
7473exp32 616 . . . . . . . 8  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( v  =  z  ->  ( u  =  w  ->  (
z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) ) ) )
7574exlimdv 1787 . . . . . . 7  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( E. v  v  =  z  ->  ( u  =  w  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) ) )
7654, 75mpi 20 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( u  =  w  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
7776exlimdv 1787 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( E. u  u  =  w  ->  ( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) ) )
7853, 77mpi 20 . . . 4  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) )
7978a1d 25 . . 3  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
8079a1d 25 . 2  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) ) )
8117, 35, 52, 804cases 964 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450   E.wex 1671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-c5 32519  ax-c4 32520  ax-c7 32521  ax-c11 32523  ax-c9 32526
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by: (None)
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