Theorem wl-equsal1i 33329

Description: The antecedent x = y is irrelevant, if one or both setvar variables are not free in ph. (Contributed by Wolf Lammen, 1-Sep-2018.)

Hypotheses

Ref	Expression
wl-equsal1i.1	|- ( F/ x ph \/ F/ y ph )
wl-equsal1i.2	|- ( x = y -> ph )

Assertion

Ref	Expression
wl-equsal1i	|- ph

Proof of Theorem wl-equsal1i

Step	Hyp	Ref	Expression
1		wl-equsal1i.1	. 2 |- ( F/ x ph \/ F/ y ph )
2		wl-equsal1i.2	. . 3 |- ( x = y -> ph )
3	2	gen2 1723	. 2 |- A. x A. y ( x = y -> ph )
4		sp 2053	. . . . 5 |- ( A. y A. x ( x = y -> ph ) -> A. x ( x = y -> ph ) )
5	4	alcoms 2035	. . . 4 |- ( A. x A. y ( x = y -> ph ) -> A. x ( x = y -> ph ) )
6		wl-equsal1t 33327	. . . 4 |- ( F/ x ph -> ( A. x ( x = y -> ph ) <-> ph ) )
7	5, 6	syl5ib 234	. . 3 |- ( F/ x ph -> ( A. x A. y ( x = y -> ph ) -> ph ) )
8		wl-equsalcom 33328	. . . . 5 |- ( A. y ( y = x -> ph ) <-> A. y ( x = y -> ph ) )
9		wl-equsal1t 33327	. . . . . 6 |- ( F/ y ph -> ( A. y ( y = x -> ph ) <-> ph ) )
10	9	biimpd 219	. . . . 5 |- ( F/ y ph -> ( A. y ( y = x -> ph ) -> ph ) )
11	8, 10	syl5bir 233	. . . 4 |- ( F/ y ph -> ( A. y ( x = y -> ph ) -> ph ) )
12	11	spsd 2057	. . 3 |- ( F/ y ph -> ( A. x A. y ( x = y -> ph ) -> ph ) )
13	7, 12	jaoi 394	. 2 |- ( ( F/ x ph \/ F/ y ph ) -> ( A. x A. y ( x = y -> ph ) -> ph ) )
14	1, 3, 13	mp2 9	1 |- ph

Syntax hints:   -> wi 4   \/ wo 383   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-11 2034  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: (None)