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Theorem equs5eOLD 1900
Description: Obsolete proof of equs5e 1899 as of 15-Jan-2018. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
equs5eOLD  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )

Proof of Theorem equs5eOLD
StepHypRef Expression
1 nfe1 1739 . 2  |-  F/ x E. x ( x  =  y  /\  ph )
2 equs3 1651 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  -. 
A. x ( x  =  y  ->  -.  ph ) )
3 ax-11 1753 . . . . 5  |-  ( x  =  y  ->  ( A. y  -.  ph  ->  A. x ( x  =  y  ->  -.  ph )
) )
43con3rr3 130 . . . 4  |-  ( -. 
A. x ( x  =  y  ->  -.  ph )  ->  ( x  =  y  ->  -.  A. y  -.  ph ) )
5 df-ex 1548 . . . 4  |-  ( E. y ph  <->  -.  A. y  -.  ph )
64, 5syl6ibr 219 . . 3  |-  ( -. 
A. x ( x  =  y  ->  -.  ph )  ->  ( x  =  y  ->  E. y ph ) )
72, 6sylbi 188 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  ( x  =  y  ->  E. y ph )
)
81, 7alrimi 1773 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 359   A.wal 1546   E.wex 1547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
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