Theorem wl-equsald 33325

Description: Deduction version of equsal 2291. (Contributed by Wolf Lammen, 27-Jul-2019.)

Hypotheses

Ref	Expression
wl-equsald.1	|- F/ x ph
wl-equsald.2	|- ( ph -> F/ x ch )
wl-equsald.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
wl-equsald	|- ( ph -> ( A. x ( x = y -> ps ) <-> ch ) )

Proof of Theorem wl-equsald

Step	Hyp	Ref	Expression
1		wl-equsald.2	. . 3 |- ( ph -> F/ x ch )
2		19.23t 2079	. . 3 |- ( F/ x ch -> ( A. x ( x = y -> ch ) <-> ( E. x x = y -> ch ) ) )
3	1, 2	syl 17	. 2 |- ( ph -> ( A. x ( x = y -> ch ) <-> ( E. x x = y -> ch ) ) )
4		wl-equsald.1	. . 3 |- F/ x ph
5		wl-equsald.3	. . . 4 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
6	5	pm5.74d 262	. . 3 |- ( ph -> ( ( x = y -> ps ) <-> ( x = y -> ch ) ) )
7	4, 6	albid 2090	. 2 |- ( ph -> ( A. x ( x = y -> ps ) <-> A. x ( x = y -> ch ) ) )
8		ax6e 2250	. . . 4 |- E. x x = y
9	8	a1bi 352	. . 3 |- ( ch <-> ( E. x x = y -> ch ) )
10	9	a1i 11	. 2 |- ( ph -> ( ch <-> ( E. x x = y -> ch ) ) )
11	3, 7, 10	3bitr4d 300	1 |- ( ph -> ( A. x ( x = y -> ps ) <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   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-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-equsal  33326  wl-equsal1t  33327  wl-sb6rft  33330