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Theorem equs5e 2069
Description: A property related to substitution that unlike equs5 2182 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.)
Assertion
Ref Expression
equs5e  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )

Proof of Theorem equs5e
StepHypRef Expression
1 nfa1 1979 . 2  |-  F/ x A. x ( x  =  y  ->  E. y ph )
2 hbe1 1917 . . . . 5  |-  ( E. y ph  ->  A. y E. y ph )
3219.23bi 1949 . . . 4  |-  ( ph  ->  A. y E. y ph )
4 ax-12 1933 . . . 4  |-  ( x  =  y  ->  ( A. y E. y ph  ->  A. x ( x  =  y  ->  E. y ph ) ) )
53, 4syl5 33 . . 3  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  E. y ph ) ) )
65imp 431 . 2  |-  ( ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  E. y ph )
)
71, 6exlimi 1995 1  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371   A.wal 1442   E.wex 1663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by:  sb4e  2088
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