Theorem cbvalw 1968

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

Hypotheses

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

Assertion

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

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. x ph -> A. y A. x ph )
2		cbvalw.2	. . 3 |- ( -. ps -> A. x -. ps )
3		cbvalw.5	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	biimpd 219	. . 3 |- ( x = y -> ( ph -> ps ) )
5	1, 2, 4	cbvaliw 1933	. 2 |- ( A. x ph -> A. y ps )
6		cbvalw.3	. . 3 |- ( A. y ps -> A. x A. y ps )
7		cbvalw.4	. . 3 |- ( -. ph -> A. y -. ph )
8	3	biimprd 238	. . . 4 |- ( x = y -> ( ps -> ph ) )
9	8	equcoms 1947	. . 3 |- ( y = x -> ( ps -> ph ) )
10	6, 7, 9	cbvaliw 1933	. 2 |- ( A. y ps -> A. x ph )
11	5, 10	impbii 199	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:  cbvalvw  1969  hbn1fw  1972