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Theorem equs5or 1758
Description: Lemma used in proofs of substitution properties. Like equs5 1757 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)
Assertion
Ref Expression
equs5or  |-  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )

Proof of Theorem equs5or
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 a9e 1631 . 2  |-  E. z 
z  =  y
2 dveeq2or 1744 . . . . . 6  |-  ( A. x  x  =  y  \/  F/ x  z  =  y )
3 nfnf1 1481 . . . . . . . . . . 11  |-  F/ x F/ x  z  =  y
43nfri 1457 . . . . . . . . . 10  |-  ( F/ x  z  =  y  ->  A. x F/ x  z  =  y )
5 ax11v 1755 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph )
) )
6 equequ2 1646 . . . . . . . . . . . . . . 15  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
76adantl 271 . . . . . . . . . . . . . 14  |-  ( ( F/ x  z  =  y  /\  z  =  y )  ->  (
x  =  z  <->  x  =  y ) )
8 nfr 1456 . . . . . . . . . . . . . . . . 17  |-  ( F/ x  z  =  y  ->  ( z  =  y  ->  A. x  z  =  y )
)
98imp 122 . . . . . . . . . . . . . . . 16  |-  ( ( F/ x  z  =  y  /\  z  =  y )  ->  A. x  z  =  y )
10 hba1 1478 . . . . . . . . . . . . . . . . 17  |-  ( A. x  z  =  y  ->  A. x A. x  z  =  y )
116imbi1d 229 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
1211sps 1475 . . . . . . . . . . . . . . . . 17  |-  ( A. x  z  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( x  =  y  ->  ph )
) )
1310, 12albidh 1414 . . . . . . . . . . . . . . . 16  |-  ( A. x  z  =  y  ->  ( A. x ( x  =  z  ->  ph )  <->  A. x ( x  =  y  ->  ph )
) )
149, 13syl 14 . . . . . . . . . . . . . . 15  |-  ( ( F/ x  z  =  y  /\  z  =  y )  ->  ( A. x ( x  =  z  ->  ph )  <->  A. x
( x  =  y  ->  ph ) ) )
1514imbi2d 228 . . . . . . . . . . . . . 14  |-  ( ( F/ x  z  =  y  /\  z  =  y )  ->  (
( ph  ->  A. x
( x  =  z  ->  ph ) )  <->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
167, 15imbi12d 232 . . . . . . . . . . . . 13  |-  ( ( F/ x  z  =  y  /\  z  =  y )  ->  (
( x  =  z  ->  ( ph  ->  A. x ( x  =  z  ->  ph ) ) )  <->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
175, 16mpbii 146 . . . . . . . . . . . 12  |-  ( ( F/ x  z  =  y  /\  z  =  y )  ->  (
x  =  y  -> 
( ph  ->  A. x
( x  =  y  ->  ph ) ) ) )
1817ex 113 . . . . . . . . . . 11  |-  ( F/ x  z  =  y  ->  ( z  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) ) )
1918imp4a 341 . . . . . . . . . 10  |-  ( F/ x  z  =  y  ->  ( z  =  y  ->  ( (
x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) ) )
204, 19alrimih 1403 . . . . . . . . 9  |-  ( F/ x  z  =  y  ->  A. x ( z  =  y  ->  (
( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph ) ) ) )
21 19.21t 1519 . . . . . . . . 9  |-  ( F/ x  z  =  y  ->  ( A. x
( z  =  y  ->  ( ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )  <->  ( z  =  y  ->  A. x
( ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) ) ) )
2220, 21mpbid 145 . . . . . . . 8  |-  ( F/ x  z  =  y  ->  ( z  =  y  ->  A. x
( ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) ) )
23 hba1 1478 . . . . . . . . 9  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x A. x ( x  =  y  ->  ph ) )
242319.23h 1432 . . . . . . . 8  |-  ( A. x ( ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
)  <->  ( E. x
( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph ) ) )
2522, 24syl6ib 159 . . . . . . 7  |-  ( F/ x  z  =  y  ->  ( z  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) ) )
2625orim2i 713 . . . . . 6  |-  ( ( A. x  x  =  y  \/  F/ x  z  =  y )  ->  ( A. x  x  =  y  \/  (
z  =  y  -> 
( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) ) ) )
272, 26ax-mp 7 . . . . 5  |-  ( A. x  x  =  y  \/  ( z  =  y  ->  ( E. x
( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph ) ) ) )
28 pm2.76 757 . . . . 5  |-  ( ( A. x  x  =  y  \/  ( z  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) ) )  -> 
( ( A. x  x  =  y  \/  z  =  y )  ->  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) ) ) )
2927, 28ax-mp 7 . . . 4  |-  ( ( A. x  x  =  y  \/  z  =  y )  ->  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) ) )
3029olcs 690 . . 3  |-  ( z  =  y  ->  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) ) )
3130exlimiv 1534 . 2  |-  ( E. z  z  =  y  ->  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) ) )
321, 31ax-mp 7 1  |-  ( A. x  x  =  y  \/  ( E. x ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664   A.wal 1287   F/wnf 1394   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  sb4or  1761
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