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Theorem ax11eq 2132
Description: Basis step for constructing a substitution instance of ax-11o 2080 without using ax-11o 2080. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax11eq  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )

Proof of Theorem ax11eq
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1580 . . 3  |-  ( A. x ( x  =  z  /\  x  =  w )  <->  ( A. x  x  =  z  /\  A. x  x  =  w ) )
2 equid 1644 . . . . . . . 8  |-  x  =  x
32a1i 10 . . . . . . 7  |-  ( x  =  y  ->  x  =  x )
43ax-gen 1533 . . . . . 6  |-  A. x
( x  =  y  ->  x  =  x )
54a1i 10 . . . . 5  |-  ( x  =  x  ->  A. x
( x  =  y  ->  x  =  x ) )
6 equequ1 1648 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  =  x  <->  z  =  x ) )
7 equequ2 1649 . . . . . . . . 9  |-  ( x  =  w  ->  (
z  =  x  <->  z  =  w ) )
86, 7sylan9bb 680 . . . . . . . 8  |-  ( ( x  =  z  /\  x  =  w )  ->  ( x  =  x  <-> 
z  =  w ) )
98sps-o 2098 . . . . . . 7  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  (
x  =  x  <->  z  =  w ) )
10 nfa1-o 2105 . . . . . . . 8  |-  F/ x A. x ( x  =  z  /\  x  =  w )
119imbi2d 307 . . . . . . . 8  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  (
( x  =  y  ->  x  =  x )  <->  ( x  =  y  ->  z  =  w ) ) )
1210, 11albid 1752 . . . . . . 7  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  ( A. x ( x  =  y  ->  x  =  x )  <->  A. x
( x  =  y  ->  z  =  w ) ) )
139, 12imbi12d 311 . . . . . 6  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  (
( x  =  x  ->  A. x ( x  =  y  ->  x  =  x ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
1413adantr 451 . . . . 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 202 . . . 4  |-  ( ( A. x ( x  =  z  /\  x  =  w )  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) )
1615exp32 588 . . 3  |-  ( A. x ( x  =  z  /\  x  =  w )  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) ) )
171, 16sylbir 204 . 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 1648 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  w  <->  y  =  w ) )
1918ad2antll 709 . . . . . 6  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( x  =  w  <-> 
y  =  w ) )
20 ax12o 1875 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  w  ->  ( y  =  w  ->  A. x  y  =  w )
) )
2120impcom 419 . . . . . . . 8  |-  ( ( -.  A. x  x  =  w  /\  -.  A. x  x  =  y )  ->  ( y  =  w  ->  A. x  y  =  w )
)
2221adantrr 697 . . . . . . 7  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( y  =  w  ->  A. x  y  =  w ) )
23 equtrr 1653 . . . . . . . 8  |-  ( y  =  w  ->  (
x  =  y  ->  x  =  w )
)
2423alimi 1546 . . . . . . 7  |-  ( A. x  y  =  w  ->  A. x ( x  =  y  ->  x  =  w ) )
2522, 24syl6 29 . . . . . 6  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( y  =  w  ->  A. x ( x  =  y  ->  x  =  w ) ) )
2619, 25sylbid 206 . . . . 5  |-  ( ( -.  A. x  x  =  w  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( x  =  w  ->  A. x ( x  =  y  ->  x  =  w ) ) )
2726adantll 694 . . . 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 1648 . . . . . . 7  |-  ( x  =  z  ->  (
x  =  w  <->  z  =  w ) )
2928sps-o 2098 . . . . . 6  |-  ( A. x  x  =  z  ->  ( x  =  w  <-> 
z  =  w ) )
3029imbi2d 307 . . . . . . 7  |-  ( A. x  x  =  z  ->  ( ( x  =  y  ->  x  =  w )  <->  ( x  =  y  ->  z  =  w ) ) )
3130dral2-o 2120 . . . . . 6  |-  ( A. x  x  =  z  ->  ( A. x ( x  =  y  ->  x  =  w )  <->  A. x ( x  =  y  ->  z  =  w ) ) )
3229, 31imbi12d 311 . . . . 5  |-  ( A. x  x  =  z  ->  ( ( x  =  w  ->  A. x
( x  =  y  ->  x  =  w ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
3332ad2antrr 706 . . . 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 201 . . 3  |-  ( ( ( A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) )
3534exp32 588 . 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 1649 . . . . . . 7  |-  ( x  =  y  ->  (
z  =  x  <->  z  =  y ) )
3736ad2antll 709 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  x  <-> 
z  =  y ) )
38 ax12o 1875 . . . . . . . . 9  |-  ( -. 
A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
) )
3938imp 418 . . . . . . . 8  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  y )  ->  ( z  =  y  ->  A. x  z  =  y )
)
4039adantrr 697 . . . . . . 7  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  y  ->  A. x  z  =  y ) )
4136biimprcd 216 . . . . . . . 8  |-  ( z  =  y  ->  (
x  =  y  -> 
z  =  x ) )
4241alimi 1546 . . . . . . 7  |-  ( A. x  z  =  y  ->  A. x ( x  =  y  ->  z  =  x ) )
4340, 42syl6 29 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  y  ->  A. x ( x  =  y  ->  z  =  x ) ) )
4437, 43sylbid 206 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  ( -.  A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  x  ->  A. x ( x  =  y  ->  z  =  x ) ) )
4544adantlr 695 . . . 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 2098 . . . . . 6  |-  ( A. x  x  =  w  ->  ( z  =  x  <-> 
z  =  w ) )
4746imbi2d 307 . . . . . . 7  |-  ( A. x  x  =  w  ->  ( ( x  =  y  ->  z  =  x )  <->  ( x  =  y  ->  z  =  w ) ) )
4847dral2-o 2120 . . . . . 6  |-  ( A. x  x  =  w  ->  ( A. x ( x  =  y  -> 
z  =  x )  <->  A. x ( x  =  y  ->  z  =  w ) ) )
4946, 48imbi12d 311 . . . . 5  |-  ( A. x  x  =  w  ->  ( ( z  =  x  ->  A. x
( x  =  y  ->  z  =  x ) )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
5049ad2antlr 707 . . . 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 201 . . 3  |-  ( ( ( -.  A. x  x  =  z  /\  A. x  x  =  w )  /\  ( -. 
A. x  x  =  y  /\  x  =  y ) )  -> 
( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) )
5251exp32 588 . 2  |-  ( ( -.  A. x  x  =  z  /\  A. x  x  =  w
)  ->  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) ) )
53 a9ev 1637 . . . . 5  |-  E. u  u  =  w
54 a9ev 1637 . . . . . . 7  |-  E. v 
v  =  z
55 ax-1 5 . . . . . . . . . . 11  |-  ( v  =  u  ->  (
x  =  y  -> 
v  =  u ) )
5655alrimiv 1617 . . . . . . . . . 10  |-  ( v  =  u  ->  A. x
( x  =  y  ->  v  =  u ) )
57 equequ1 1648 . . . . . . . . . . . . 13  |-  ( v  =  z  ->  (
v  =  u  <->  z  =  u ) )
58 equequ2 1649 . . . . . . . . . . . . 13  |-  ( u  =  w  ->  (
z  =  u  <->  z  =  w ) )
5957, 58sylan9bb 680 . . . . . . . . . . . 12  |-  ( ( v  =  z  /\  u  =  w )  ->  ( v  =  u  <-> 
z  =  w ) )
6059adantl 452 . . . . . . . . . . 11  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( v  =  u  <->  z  =  w ) )
61 dveeq2-o 2123 . . . . . . . . . . . . . . 15  |-  ( -. 
A. x  x  =  z  ->  ( v  =  z  ->  A. x  v  =  z )
)
62 dveeq2-o 2123 . . . . . . . . . . . . . . 15  |-  ( -. 
A. x  x  =  w  ->  ( u  =  w  ->  A. x  u  =  w )
)
6361, 62im2anan9 808 . . . . . . . . . . . . . 14  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( (
v  =  z  /\  u  =  w )  ->  ( A. x  v  =  z  /\  A. x  u  =  w
) ) )
6463imp 418 . . . . . . . . . . . . 13  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( A. x  v  =  z  /\  A. x  u  =  w ) )
65 19.26 1580 . . . . . . . . . . . . 13  |-  ( A. x ( v  =  z  /\  u  =  w )  <->  ( A. x  v  =  z  /\  A. x  u  =  w ) )
6664, 65sylibr 203 . . . . . . . . . . . 12  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  A. x
( v  =  z  /\  u  =  w ) )
67 nfa1-o 2105 . . . . . . . . . . . . 13  |-  F/ x A. x ( v  =  z  /\  u  =  w )
6859sps-o 2098 . . . . . . . . . . . . . 14  |-  ( A. x ( v  =  z  /\  u  =  w )  ->  (
v  =  u  <->  z  =  w ) )
6968imbi2d 307 . . . . . . . . . . . . 13  |-  ( A. x ( v  =  z  /\  u  =  w )  ->  (
( x  =  y  ->  v  =  u )  <->  ( x  =  y  ->  z  =  w ) ) )
7067, 69albid 1752 . . . . . . . . . . . 12  |-  ( A. x ( v  =  z  /\  u  =  w )  ->  ( A. x ( x  =  y  ->  v  =  u )  <->  A. x
( x  =  y  ->  z  =  w ) ) )
7166, 70syl 15 . . . . . . . . . . 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 311 . . . . . . . . . 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 202 . . . . . . . . 9  |-  ( ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  /\  (
v  =  z  /\  u  =  w )
)  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) )
7473exp32 588 . . . . . . . 8  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( v  =  z  ->  ( u  =  w  ->  (
z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) ) ) )
7574exlimdv 1664 . . . . . . 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 16 . . . . . 6  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( u  =  w  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
7776exlimdv 1664 . . . . 5  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( E. u  u  =  w  ->  ( z  =  w  ->  A. x ( x  =  y  ->  z  =  w ) ) ) )
7853, 77mpi 16 . . . 4  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) )
7978a1d 22 . . 3  |-  ( ( -.  A. x  x  =  z  /\  -.  A. x  x  =  w )  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
8079a1d 22 . 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 915 1  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( z  =  w  ->  A. x
( x  =  y  ->  z  =  w ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-4 2074  ax-5o 2075  ax-6o 2076  ax-10o 2078  ax-12o 2081
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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