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Theorem ax12eq 2242
Description: Basis step for constructing a substitution instance of ax-c15 2191 without using ax-c15 2191. 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 1647 . . 3  |-  ( A. x ( x  =  z  /\  x  =  w )  <->  ( A. x  x  =  z  /\  A. x  x  =  w ) )
2 equid 1729 . . . . . . . 8  |-  x  =  x
32a1i 11 . . . . . . 7  |-  ( x  =  y  ->  x  =  x )
43ax-gen 1591 . . . . . 6  |-  A. x
( x  =  y  ->  x  =  x )
54a1i 11 . . . . 5  |-  ( x  =  x  ->  A. x
( x  =  y  ->  x  =  x ) )
6 equequ1 1736 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  =  x  <->  z  =  x ) )
7 equequ2 1737 . . . . . . . . 9  |-  ( x  =  w  ->  (
z  =  x  <->  z  =  w ) )
86, 7sylan9bb 699 . . . . . . . 8  |-  ( ( x  =  z  /\  x  =  w )  ->  ( x  =  x  <-> 
z  =  w ) )
98sps-o 2209 . . . . . . 7  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  (
x  =  x  <->  z  =  w ) )
10 nfa1-o 2216 . . . . . . . 8  |-  F/ x A. x ( x  =  z  /\  x  =  w )
119imbi2d 316 . . . . . . . 8  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  (
( x  =  y  ->  x  =  x )  <->  ( x  =  y  ->  z  =  w ) ) )
1210, 11albid 1819 . . . . . . 7  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  ( A. x ( x  =  y  ->  x  =  x )  <->  A. x
( x  =  y  ->  z  =  w ) ) )
139, 12imbi12d 320 . . . . . 6  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  (
( x  =  x  ->  A. x ( x  =  y  ->  x  =  x ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
1413adantr 465 . . . . 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 211 . . . 4  |-  ( ( A. x ( x  =  z  /\  x  =  w )  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) )
1615exp32 605 . . 3  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) ) )
171, 16sylbir 213 . 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 1736 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  w  <->  y  =  w ) )
1918ad2antll 728 . . . . . 6  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( x  =  w  <-> 
y  =  w ) )
20 axc9 1994 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  w  ->  ( y  =  w  ->  A. x  y  =  w )
) )
2120impcom 430 . . . . . . . 8  |-  ( ( -.  A. x  x  =  w  /\  -.  A. x  x  =  y )  ->  ( y  =  w  ->  A. x  y  =  w )
)
2221adantrr 716 . . . . . . 7  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( y  =  w  ->  A. x  y  =  w ) )
23 equtrr 1735 . . . . . . . 8  |-  ( y  =  w  ->  (
x  =  y  ->  x  =  w )
)
2423alimi 1604 . . . . . . 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 215 . . . . 5  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( x  =  w  ->  A. x ( x  =  y  ->  x  =  w ) ) )
2726adantll 713 . . . 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 1736 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  w  <->  z  =  w ) )
2928sps-o 2209 . . . . . 6  |-  ( A. x  x  =  z  ->  ( x  =  w  <-> 
z  =  w ) )
3029imbi2d 316 . . . . . . 7  |-  ( A. x  x  =  z  ->  ( ( x  =  y  ->  x  =  w )  <->  ( x  =  y  ->  z  =  w ) ) )
3130dral2-o 2231 . . . . . 6  |-  ( A. x  x  =  z  ->  ( A. x ( x  =  y  ->  x  =  w )  <->  A. x ( x  =  y  ->  z  =  w ) ) )
3229, 31imbi12d 320 . . . . 5  |-  ( A. x  x  =  z  ->  ( ( x  =  w  ->  A. x
( x  =  y  ->  x  =  w ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
3332ad2antrr 725 . . . 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 210 . . 3  |-  ( ( ( A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) )
3534exp32 605 . 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 1737 . . . . . . 7  |-  ( x  =  y  ->  (
z  =  x  <->  z  =  y ) )
3736ad2antll 728 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  x  <-> 
z  =  y ) )
38 axc9 1994 . . . . . . . . 9  |-  ( -. 
A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
) )
3938imp 429 . . . . . . . 8  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( z  =  y  ->  A. x  z  =  y )
)
4039adantrr 716 . . . . . . 7  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  y  ->  A. x  z  =  y ) )
4136biimprcd 225 . . . . . . . 8  |-  ( z  =  y  ->  (
x  =  y  -> 
z  =  x ) )
4241alimi 1604 . . . . . . 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 215 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  x  ->  A. x ( x  =  y  ->  z  =  x ) ) )
4544adantlr 714 . . . 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 2209 . . . . . 6  |-  ( A. x  x  =  w  ->  ( z  =  x  <-> 
z  =  w ) )
4746imbi2d 316 . . . . . . 7  |-  ( A. x  x  =  w  ->  ( ( x  =  y  ->  z  =  x )  <->  ( x  =  y  ->  z  =  w ) ) )
4847dral2-o 2231 . . . . . 6  |-  ( A. x  x  =  w  ->  ( A. x ( x  =  y  -> 
z  =  x )  <->  A. x ( x  =  y  ->  z  =  w ) ) )
4946, 48imbi12d 320 . . . . 5  |-  ( A. x  x  =  w  ->  ( ( z  =  x  ->  A. x
( x  =  y  ->  z  =  x ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
5049ad2antlr 726 . . . 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 210 . . 3  |-  ( ( ( -.  A. x  x  =  z  /\  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) )
5251exp32 605 . 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 1710 . . . . 5  |-  E. u  u  =  w
54 ax6ev 1710 . . . . . . 7  |-  E. v 
v  =  z
55 ax-1 6 . . . . . . . . . . 11  |-  ( v  =  u  ->  (
x  =  y  -> 
v  =  u ) )
5655alrimiv 1685 . . . . . . . . . 10  |-  ( v  =  u  ->  A. x
( x  =  y  ->  v  =  u ) )
57 equequ1 1736 . . . . . . . . . . . . 13  |-  ( v  =  z  ->  (
v  =  u  <->  z  =  u ) )
58 equequ2 1737 . . . . . . . . . . . . 13  |-  ( u  =  w  ->  (
z  =  u  <->  z  =  w ) )
5957, 58sylan9bb 699 . . . . . . . . . . . 12  |-  ( ( v  =  z  /\  u  =  w )  ->  ( v  =  u  <-> 
z  =  w ) )
6059adantl 466 . . . . . . . . . . 11  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( v  =  u  <->  z  =  w ) )
61 dveeq2-o 2234 . . . . . . . . . . . . . . 15  |-  ( -. 
A. x  x  =  z  ->  ( v  =  z  ->  A. x  v  =  z )
)
62 dveeq2-o 2234 . . . . . . . . . . . . . . 15  |-  ( -. 
A. x  x  =  w  ->  ( u  =  w  ->  A. x  u  =  w )
)
6361, 62im2anan9 831 . . . . . . . . . . . . . 14  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( (
v  =  z  /\  u  =  w )  ->  ( A. x  v  =  z  /\  A. x  u  =  w
) ) )
6463imp 429 . . . . . . . . . . . . 13  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( A. x  v  =  z  /\  A. x  u  =  w ) )
65 19.26 1647 . . . . . . . . . . . . 13  |-  ( A. x ( v  =  z  /\  u  =  w )  <->  ( A. x  v  =  z  /\  A. x  u  =  w ) )
6664, 65sylibr 212 . . . . . . . . . . . 12  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  A. x
( v  =  z  /\  u  =  w ) )
67 nfa1-o 2216 . . . . . . . . . . . . 13  |-  F/ x A. x ( v  =  z  /\  u  =  w )
6859sps-o 2209 . . . . . . . . . . . . . 14  |-  ( A. x ( v  =  z  /\  u  =  w )  ->  (
v  =  u  <->  z  =  w ) )
6968imbi2d 316 . . . . . . . . . . . . 13  |-  ( A. x ( v  =  z  /\  u  =  w )  ->  (
( x  =  y  ->  v  =  u )  <->  ( x  =  y  ->  z  =  w ) ) )
7067, 69albid 1819 . . . . . . . . . . . 12  |-  ( A. x ( v  =  z  /\  u  =  w )  ->  ( A. x ( x  =  y  ->  v  =  u )  <->  A. x
( x  =  y  ->  z  =  w ) ) )
7166, 70syl 16 . . . . . . . . . . 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 320 . . . . . . . . . 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 211 . . . . . . . . 9  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) )
7473exp32 605 . . . . . . . 8  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( v  =  z  ->  ( u  =  w  ->  (
z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) ) ) )
7574exlimdv 1690 . . . . . . 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 17 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( u  =  w  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
7776exlimdv 1690 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( E. u  u  =  w  ->  ( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) ) )
7853, 77mpi 17 . . . 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 940 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 184    /\ wa 369   A.wal 1367   E.wex 1586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-c5 2185  ax-c4 2186  ax-c7 2187  ax-c11 2189  ax-c9 2192
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590
This theorem is referenced by: (None)
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