Theorem wl-equsal1t 33327

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 2291, spimt 2253 or sbft 2379, 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

Step	Hyp	Ref	Expression
1		nfnf1 2031	. 2 |- F/ x F/ x ph
2		id 22	. 2 |- ( F/ x ph -> F/ x ph )
3		biid 251	. . 3 |- ( ph <-> ph )
4	3	2a1i 12	. 2 |- ( F/ x ph -> ( x = y -> ( ph <-> ph ) ) )
5	1, 2, 4	wl-equsald 33325	1 |- ( F/ x ph -> ( A. x ( x = y -> ph ) <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  wl-equsal1i  33329