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Theorem equs4 1944
Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
Assertion
Ref Expression
equs4  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)

Proof of Theorem equs4
StepHypRef Expression
1 a9e 1941 . 2  |-  E. x  x  =  y
2 exintr 1621 . 2  |-  ( A. x ( x  =  y  ->  ph )  -> 
( E. x  x  =  y  ->  E. x
( x  =  y  /\  ph ) ) )
31, 2mpi 17 1  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547
This theorem is referenced by:  equsex  1962  equs45f  2029  sb2  2058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548
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