Theorem equsalhwOLD 1839

Description: Obsolete proof of equsalhw 1838 as of 28-Dec-2017. (Contributed by NM, 29-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable group:   x,y

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

Proof of Theorem equsalhwOLD

Step	Hyp	Ref	Expression
1		equsalhwOLD.2	. . . . 5 |- (x = y -> (ph <-> ps))
2		sp 1747	. . . . . 6 |- (A.xps -> ps)
3		equsalhwOLD.1	. . . . . 6 |- (ps -> A.xps)
4	2, 3	impbii 180	. . . . 5 |- (A.xps <-> ps)
5	1, 4	syl6bbr 254	. . . 4 |- (x = y -> (ph <-> A.xps))
6	5	pm5.74i 236	. . 3 |- ((x = y -> ph) <-> (x = y -> A.xps))
7	6	albii 1566	. 2 |- (A.x(x = y -> ph) <-> A.x(x = y -> A.xps))
8	3	a1d 22	. . . 4 |- (ps -> (x = y -> A.xps))
9	3, 8	alrimih 1565	. . 3 |- (ps -> A.x(x = y -> A.xps))
10		ax9v 1655	. . . . 5 |- -. A.x -. x = y
11		con3 126	. . . . . 6 |- ((x = y -> A.xps) -> (-. A.xps -> -. x = y))
12	11	al2imi 1561	. . . . 5 |- (A.x(x = y -> A.xps) -> (A.x -. A.xps -> A.x -. x = y))
13	10, 12	mtoi 169	. . . 4 |- (A.x(x = y -> A.xps) -> -. A.x -. A.xps)
14		ax6o 1750	. . . 4 |- (-. A.x -. A.xps -> ps)
15	13, 14	syl 15	. . 3 |- (A.x(x = y -> A.xps) -> ps)
16	9, 15	impbii 180	. 2 |- (ps <-> A.x(x = y -> A.xps))
17	7, 16	bitr4i 243	1 |- (A.x(x = y -> ph) <-> ps)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540

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

This theorem is referenced by: (None)