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Theorem equs5e 1775
Description: A property related to substitution that unlike equs5 1809 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
equs5e  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )

Proof of Theorem equs5e
StepHypRef Expression
1 19.8a 1570 . . . . 5  |-  ( ph  ->  E. y ph )
2 hbe1 1475 . . . . 5  |-  ( E. y ph  ->  A. y E. y ph )
31, 2syl 14 . . . 4  |-  ( ph  ->  A. y E. y ph )
43anim2i 340 . . 3  |-  ( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  A. y E. y ph ) )
54eximi 1580 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  A. y E. y ph )
)
6 equs5a 1774 . 2  |-  ( E. x ( x  =  y  /\  A. y E. y ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
75, 6syl 14 1  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1333    = wceq 1335   E.wex 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-11 1486  ax-4 1490  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  ax11e  1776  sb4e  1785
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