Theorem equsalhw 2123

Description: Weaker version of equsalh 2294 with a dv condition which does not require ax-13 2246. (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 2122	. 2 |- ( A. x ( x = y -> ps ) <-> ( E. x x = y -> ps ) )
3		equsalhw.2	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	pm5.74i 260	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> ps ) )
5	4	albii 1747	. 2 |- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> ps ) )
6		ax6ev 1890	. . 3 |- E. x x = y
7	6	a1bi 352	. 2 |- ( ps <-> ( E. x x = y -> ps ) )
8	2, 5, 7	3bitr4i 292	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704

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

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

This theorem is referenced by:  dvelimhw  2173