Theorem cbvalvw 1702

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

Hypothesis

Ref	Expression
cbvalvw.1	|- (x = y -> (ph <-> ps))

Assertion

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

Distinct variable groups:   x,y   ps,x   ph,y

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

Proof of Theorem cbvalvw

Step	Hyp	Ref	Expression
1		cbvalvw.1	. . . 4 |- (x = y -> (ph <-> ps))
2	1	biimpd 198	. . 3 |- (x = y -> (ph -> ps))
3	2	cbvalivw 1674	. 2 |- (A.xph -> A.yps)
4	1	biimprd 214	. . . 4 |- (x = y -> (ps -> ph))
5	4	equcoms 1681	. . 3 |- (y = x -> (ps -> ph))
6	5	cbvalivw 1674	. 2 |- (A.yps -> A.xph)
7	3, 6	impbii 180	1 |- (A.xph <-> A.yps)

Syntax hints:   -> 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:  cbvexvw  1703  hba1w  1707  ax11wdemo  1723