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Theorem equs4 1725
Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
Assertion
Ref Expression
equs4  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)

Proof of Theorem equs4
StepHypRef Expression
1 a9e 1696 . . 3  |-  E. x  x  =  y
2 19.29 1620 . . 3  |-  ( ( A. x ( x  =  y  ->  ph )  /\  E. x  x  =  y )  ->  E. x
( ( x  =  y  ->  ph )  /\  x  =  y )
)
31, 2mpan2 425 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( ( x  =  y  ->  ph )  /\  x  =  y
) )
4 ancl 318 . . . 4  |-  ( ( x  =  y  ->  ph )  ->  ( x  =  y  ->  (
x  =  y  /\  ph ) ) )
54imp 124 . . 3  |-  ( ( ( x  =  y  ->  ph )  /\  x  =  y )  -> 
( x  =  y  /\  ph ) )
65eximi 1600 . 2  |-  ( E. x ( ( x  =  y  ->  ph )  /\  x  =  y
)  ->  E. x
( x  =  y  /\  ph ) )
73, 6syl 14 1  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1351    = wceq 1353   E.wex 1492
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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  sb2  1767  equs45f  1802
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