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Theorem equs5e 1829
Description: A property related to substitution that unlike equs5 1936 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
equs5e  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )

Proof of Theorem equs5e
StepHypRef Expression
1 nfe1 1706 . 2  |-  F/ x E. x ( x  =  y  /\  ph )
2 equs3 1625 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  -. 
A. x ( x  =  y  ->  -.  ph ) )
3 ax-11 1715 . . . . 5  |-  ( x  =  y  ->  ( A. y  -.  ph  ->  A. x ( x  =  y  ->  -.  ph )
) )
43con3rr3 128 . . . 4  |-  ( -. 
A. x ( x  =  y  ->  -.  ph )  ->  ( x  =  y  ->  -.  A. y  -.  ph ) )
5 df-ex 1529 . . . 4  |-  ( E. y ph  <->  -.  A. y  -.  ph )
64, 5syl6ibr 218 . . 3  |-  ( -. 
A. x ( x  =  y  ->  -.  ph )  ->  ( x  =  y  ->  E. y ph ) )
72, 6sylbi 187 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  ( x  =  y  ->  E. y ph )
)
81, 7alrimi 1745 1  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. 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:  sb4e  1865
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-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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