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Theorem equs4 1912
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 1904 . . 3  |-  E. x  x  =  y
2 19.29 1586 . . 3  |-  ( ( A. x ( x  =  y  ->  ph )  /\  E. x  x  =  y )  ->  E. x
( ( x  =  y  ->  ph )  /\  x  =  y )
)
31, 2mpan2 652 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( ( x  =  y  ->  ph )  /\  x  =  y
) )
4 ancl 529 . . . 4  |-  ( ( x  =  y  ->  ph )  ->  ( x  =  y  ->  (
x  =  y  /\  ph ) ) )
54imp 418 . . 3  |-  ( ( ( x  =  y  ->  ph )  /\  x  =  y )  -> 
( x  =  y  /\  ph ) )
65eximi 1566 . 2  |-  ( E. x ( ( x  =  y  ->  ph )  /\  x  =  y
)  ->  E. x
( x  =  y  /\  ph ) )
73, 6syl 15 1  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531
This theorem is referenced by:  equs45f  1942  sb2  1976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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