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Theorem equs5e 1910
Description: A property related to substitution that unlike equs5 2089 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 1806 . 2  |-  F/ x A. x ( x  =  y  ->  E. y ph )
2 hbe1 1746 . . . . 5  |-  ( E. y ph  ->  A. y E. y ph )
3219.23bi 1775 . . . 4  |-  ( ph  ->  A. y E. y ph )
4 ax-11 1761 . . . 4  |-  ( x  =  y  ->  ( A. y E. y ph  ->  A. x ( x  =  y  ->  E. y ph ) ) )
53, 4syl5 30 . . 3  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  E. y ph ) ) )
65imp 419 . 2  |-  ( ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  E. y ph )
)
71, 6exlimi 1821 1  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550
This theorem is referenced by:  sb4e  1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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