Theorem cbvalw 1701

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypotheses

Ref	Expression
cbvalw.1	|- (A.xph -> A.yA.xph)
cbvalw.2	|- (-. ps -> A.x -. ps)
cbvalw.3	|- (A.yps -> A.xA.yps)
cbvalw.4	|- (-. ph -> A.y -. ph)
cbvalw.5	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvalw	|- (A.xph <-> A.yps)

Distinct variable group:   x,y

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

Proof of Theorem cbvalw

Step	Hyp	Ref	Expression
1		cbvalw.1	. . 3 |- (A.xph -> A.yA.xph)
2		cbvalw.2	. . 3 |- (-. ps -> A.x -. ps)
3		cbvalw.5	. . . 4 |- (x = y -> (ph <-> ps))
4	3	biimpd 198	. . 3 |- (x = y -> (ph -> ps))
5	1, 2, 4	cbvaliw 1673	. 2 |- (A.xph -> A.yps)
6		cbvalw.3	. . 3 |- (A.yps -> A.xA.yps)
7		cbvalw.4	. . 3 |- (-. ph -> A.y -. ph)
8	3	biimprd 214	. . . 4 |- (x = y -> (ps -> ph))
9	8	equcoms 1681	. . 3 |- (y = x -> (ps -> ph))
10	6, 7, 9	cbvaliw 1673	. 2 |- (A.yps -> A.xph)
11	5, 10	impbii 180	1 |- (A.xph <-> A.yps)

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  hbn1fw  1705