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Theorem equsalh 1630
Description: A useful equivalence related to substitution. New proofs should use equsal 1631 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
equsalh.1  |-  ( ps 
->  A. x ps )
equsalh.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsalh  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )

Proof of Theorem equsalh
StepHypRef Expression
1 equsalh.2 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2 equsalh.1 . . . . . 6  |-  ( ps 
->  A. x ps )
3219.3h 1461 . . . . 5  |-  ( A. x ps  <->  ps )
41, 3syl6bbr 191 . . . 4  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ps )
)
54pm5.74i 173 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  A. x ps ) )
65albii 1375 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  A. x ps )
)
72a1d 22 . . . 4  |-  ( ps 
->  ( x  =  y  ->  A. x ps )
)
82, 7alrimih 1374 . . 3  |-  ( ps 
->  A. x ( x  =  y  ->  A. x ps ) )
9 ax9o 1604 . . 3  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ps )
108, 9impbii 121 . 2  |-  ( ps  <->  A. x ( x  =  y  ->  A. x ps ) )
116, 10bitr4i 180 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  sb6x  1678  dvelimfALT2  1714  dvelimALT  1902  dvelimfv  1903  dvelimor  1910
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