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Theorem wl-equsal1t 31838
Description: The expression  x  =  y in antecedent position plays an important role in predicate logic, namely in implicit substitution. However, occasionally it is irrelevant, and can safely be dropped. A sufficient condition for this is when  x (or  y or both) is not free in  ph.

This theorem is more fundamental than equsal 2093, spimt 2063 or sbft 2177, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)

Assertion
Ref Expression
wl-equsal1t  |-  ( F/ x ph  ->  ( A. x ( x  =  y  ->  ph )  <->  ph ) )

Proof of Theorem wl-equsal1t
StepHypRef Expression
1 nfnf1 1958 . 2  |-  F/ x F/ x ph
2 id 22 . 2  |-  ( F/ x ph  ->  F/ x ph )
3 biid 239 . . 3  |-  ( ph  <->  ph )
432a1i 12 . 2  |-  ( F/ x ph  ->  (
x  =  y  -> 
( ph  <->  ph ) ) )
51, 2, 4wl-equsald 31836 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   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:  wl-equsal1i  31840
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