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Theorem equs5 1938
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 1898 . 2  |-  F/ x  -.  A. x  x  =  y
2 nfa1 1758 . 2  |-  F/ x A. x ( x  =  y  ->  ph )
3 ax11o 1936 . . 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 1805 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 1529   E.wex 1530
This theorem is referenced by:  sb3  1994  sb4  1995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534
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