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Theorem equsalh 1737
Description: A useful equivalence related to substitution. New proofs should use equsal 1738 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 1564 . . . . 5  |-  ( A. x ps  <->  ps )
41, 3bitr4di 198 . . . 4  |-  ( x  =  y  ->  ( ph 
<-> 
A. x ps )
)
54pm5.74i 180 . . 3  |-  ( ( x  =  y  ->  ph )  <->  ( x  =  y  ->  A. x ps ) )
65albii 1481 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  A. x
( x  =  y  ->  A. x ps )
)
72a1d 22 . . . 4  |-  ( ps 
->  ( x  =  y  ->  A. x ps )
)
82, 7alrimih 1480 . . 3  |-  ( ps 
->  A. x ( x  =  y  ->  A. x ps ) )
9 ax9o 1709 . . 3  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ps )
108, 9impbii 126 . 2  |-  ( ps  <->  A. x ( x  =  y  ->  A. x ps ) )
116, 10bitr4i 187 1  |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  sb6x  1790  dvelimfALT2  1828  dvelimALT  2022  dvelimfv  2023  dvelimor  2030
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