Theorem cbvalvw 1969

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)

Hypothesis

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

Assertion

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

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		ax-5 1839	. 2 |- ( A. x ph -> A. y A. x ph )
2		ax-5 1839	. 2 |- ( -. ps -> A. x -. ps )
3		ax-5 1839	. 2 |- ( A. y ps -> A. x A. y ps )
4		ax-5 1839	. 2 |- ( -. ph -> A. y -. ph )
5		cbvalvw.1	. 2 |- ( x = y -> ( ph <-> ps ) )
6	1, 2, 3, 4, 5	cbvalw 1968	1 |- ( A. x ph <-> A. y ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   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-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  cbvexvw  1970  hba1w  1974  hba1wOLD  1975  ax12wdemo  2012  bj-ssbjust  32618