Theorem cbvaldva 2281

Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 18-Jul-2021.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem cbvaldva

Step	Hyp	Ref	Expression
1		cbvaldva.1	. . . . . 6 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2	1	expcom 451	. . . . 5 |- ( x = y -> ( ph -> ( ps <-> ch ) ) )
3	2	pm5.74d 262	. . . 4 |- ( x = y -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
4	3	cbvalv 2273	. . 3 |- ( A. x ( ph -> ps ) <-> A. y ( ph -> ch ) )
5		19.21v 1868	. . 3 |- ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) )
6		19.21v 1868	. . 3 |- ( A. y ( ph -> ch ) <-> ( ph -> A. y ch ) )
7	4, 5, 6	3bitr3i 290	. 2 |- ( ( ph -> A. x ps ) <-> ( ph -> A. y ch ) )
8	7	pm5.74ri 261	1 |- ( ph -> ( A. x ps <-> A. y ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   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  ax-11 2034  ax-12 2047  ax-13 2246

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

This theorem is referenced by:  cbvexdva  2283  cbval2v  2285  cbvraldva2  3175