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Theorem bj-equsal1t 31394
Description: Duplication of wl-equsal1t 31838, with shorter proof. Note: wl-equsalcom 31839 is also interesting. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-equsal1t  |-  ( F/ x ph  ->  ( A. x ( x  =  y  ->  ph )  <->  ph ) )

Proof of Theorem bj-equsal1t
StepHypRef Expression
1 bj-alequex 31268 . . 3  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ph )
2 19.9t 1946 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
31, 2syl5ib 222 . 2  |-  ( F/ x ph  ->  ( A. x ( x  =  y  ->  ph )  ->  ph ) )
4 nfr 1928 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
5 ala1 1704 . . 3  |-  ( A. x ph  ->  A. x
( x  =  y  ->  ph ) )
64, 5syl6 34 . 2  |-  ( F/ x ph  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
73, 6impbid 193 1  |-  ( F/ x ph  ->  ( A. x ( x  =  y  ->  ph )  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435   E.wex 1657   F/wnf 1661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  bj-equsal1ti  31395
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