Theorem cbvaldvaw 1945

Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 1943 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)

Hypothesis

Ref	Expression
cbvaldvaw.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )

Assertion

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

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

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

Proof of Theorem cbvaldvaw

Step	Hyp	Ref	Expression
1		cbvaldvaw.1	. . . . . 6 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2	1	ancoms 268	. . . . 5 |- ( ( x = y /\ ph ) -> ( ps <-> ch ) )
3	2	pm5.74da 443	. . . 4 |- ( x = y -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
4	3	cbvalvw 1934	. . 3 |- ( A. x ( ph -> ps ) <-> A. y ( ph -> ch ) )
5		19.21v 1887	. . 3 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
6		19.21v 1887	. . 3 |- ( A. y ( ph -> ch ) <-> ( ph -> A. y ch ) )
7	4, 5, 6	3bitr3i 210	. 2 |- ( ( ph -> A. x ps ) <-> ( ph -> A. y ch ) )
8	7	pm5.74ri 181	1 |- ( ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  cbval2vw  1947