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Theorem equs5 1936
Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
equs5  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem equs5
StepHypRef Expression
1 nfnae 1896 . 2  |-  F/ x  -.  A. x  x  =  y
2 nfa1 1756 . 2  |-  F/ x A. x ( x  =  y  ->  ph )
3 ax11o 1934 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
43imp3a 420 . 2  |-  ( -. 
A. x  x  =  y  ->  ( (
x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
51, 2, 4exlimd 1803 1  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  sb3  1992  sb4  1993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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