Theorem equsalhw 2038

Description: Weaker version of equsalh 2139 (requiring distinct variables) without using ax-13 2101. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)

Hypotheses

Ref	Expression
equsalhw.1	|- ( ps -> A. x ps )
equsalhw.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
equsalhw	|- ( A. x ( x = y -> ph ) <-> ps )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem equsalhw

Step	Hyp	Ref	Expression
1		equsalhw.1	. . 3 |- ( ps -> A. x ps )
2	1	19.23h 2004	. 2 |- ( A. x ( x = y -> ps ) <-> ( E. x x = y -> ps ) )
3		equsalhw.2	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	pm5.74i 253	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> ps ) )
5	4	albii 1701	. 2 |- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> ps ) )
6		ax6ev 1817	. . 3 |- E. x x = y
7	6	a1bi 343	. 2 |- ( ps <-> ( E. x x = y -> ps ) )
8	2, 5, 7	3bitr4i 285	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1452   E.wex 1673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678

This theorem is referenced by:  dvelimhw  2070