Theorem cbvaliw 1673

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)

Hypotheses

Ref	Expression
cbvaliw.1	|- (A.xph -> A.yA.xph)
cbvaliw.2	|- (-. ps -> A.x -. ps)
cbvaliw.3	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
cbvaliw	|- (A.xph -> A.yps)

Distinct variable group:   x,y

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

Proof of Theorem cbvaliw

Step	Hyp	Ref	Expression
1		cbvaliw.1	. 2 |- (A.xph -> A.yA.xph)
2		cbvaliw.2	. . 3 |- (-. ps -> A.x -. ps)
3		cbvaliw.3	. . 3 |- (x = y -> (ph -> ps))
4	2, 3	spimw 1668	. 2 |- (A.xph -> ps)
5	1, 4	alrimih 1565	1 |- (A.xph -> A.yps)

This proof depends on syntax axioms:  -. wn 3   -> wi 4  A.wal 1540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-9 1654

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

This theorem is used by:  cbvalw  1701