Theorem cbvaliw 1933

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. x ph -> A. y A. x ph )
cbvaliw.2	|- ( -. ps -> A. x -. ps )
cbvaliw.3	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
cbvaliw	|- ( A. x ph -> A. y ps )

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. x ph -> A. y A. x ph )
2		cbvaliw.2	. . 3 |- ( -. ps -> A. x -. ps )
3		cbvaliw.3	. . 3 |- ( x = y -> ( ph -> ps ) )
4	2, 3	spimw 1926	. 2 |- ( A. x ph -> ps )
5	1, 4	alrimih 1751	1 |- ( A. x ph -> A. y ps )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  spfw  1965  cbvalw  1968