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Theorem equs5 2149
Description: Lemma used in proofs of substitution properties. (Contributed by NM, 14-May-1993.) (Revised by BJ, 1-Oct-2018.)
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 nfna1 1962 . . 3  |-  F/ x  -.  A. x  x  =  y
2 nfa1 1956 . . 3  |-  F/ x A. x ( x  =  y  ->  ph )
3 axc15 2144 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
43impd 432 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( (
x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
51, 2, 4exlimd 1974 . 2  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )
6 equs4 2092 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
75, 6impbid1 206 1  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  sb3  2153  sb4  2154
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