Theorem cbvals 17054

Description: Rule used to change bound variables, using implicit substitution. (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

Ref	Expression
cbvals.1	|- ( x = y -> ( ph <-> ch ) )
cbvals.2	|- ( x = y -> ( ps <-> th ) )

Assertion

Ref	Expression
cbvals	|- ( A.E. x ( ph -> ps ) <-> A.E. y ( ch -> th ) )

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

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

Proof of Theorem cbvals

Step	Hyp	Ref	Expression
1		cbvals.1	. . . . 5 |- ( x = y -> ( ph <-> ch ) )
2		cbvals.2	. . . . 5 |- ( x = y -> ( ps <-> th ) )
3	1, 2	imbi12d 234	. . . 4 |- ( x = y -> ( ( ph -> ps ) <-> ( ch -> th ) ) )
4	3	cbvalvw 1975	. . 3 |- ( A. x ( ph -> ps ) <-> A. y ( ch -> th ) )
5	1	cbvexvw 1976	. . 3 |- ( E. x ph <-> E. y ch )
6	4, 5	anbi12i 464	. 2 |- ( ( A. x ( ph -> ps ) /\ E. x ph ) <-> ( A. y ( ch -> th ) /\ E. y ch ) )
7		df-als 17036	. 2 |- ( A.E. x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) )
8		df-als 17036	. 2 |- ( A.E. y ( ch -> th ) <-> ( A. y ( ch -> th ) /\ E. y ch ) )
9	6, 7, 8	3bitr4i 212	1 |- ( A.E. x ( ph -> ps ) <-> A.E. y ( ch -> th ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   E.wex 1545   A.E.wals 17034

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-als 17036

This theorem is referenced by: (None)