Theorem equsalhw 1838

Description: Weaker version of equsalh 1961 (requiring distinct variables) without using ax-12 1925. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)

Hypotheses

Ref	Expression
equsalhw.1	|- (ps -> A.xps)
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.xps)
2	1	19.23h 1802	. 2 |- (A.x(x = y -> ps) <-> (E.x x = y -> ps))
3		equsalhw.2	. . . 4 |- (x = y -> (ph <-> ps))
4	3	pm5.74i 236	. . 3 |- ((x = y -> ph) <-> (x = y -> ps))
5	4	albii 1566	. 2 |- (A.x(x = y -> ph) <-> A.x(x = y -> ps))
6		a9ev 1656	. . 3 |- E.x x = y
7	6	a1bi 327	. 2 |- (ps <-> (E.x x = y -> ps))
8	2, 5, 7	3bitr4i 268	1 |- (A.x(x = y -> ph) <-> ps)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  dvelimhw  1849