Theorem alsbii 17049

Description: Congruence: equivalents may be substituted inside an "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

Ref	Expression
alsbii.1	|- ( ph <-> ch )
alsbii.2	|- ( ps <-> th )

Assertion

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

Proof of Theorem alsbii

Step	Hyp	Ref	Expression
1		alsbii.1	. . . . 5 |- ( ph <-> ch )
2		alsbii.2	. . . . 5 |- ( ps <-> th )
3	1, 2	imbi12i 239	. . . 4 |- ( ( ph -> ps ) <-> ( ch -> th ) )
4	3	albii 1523	. . 3 |- ( A. x ( ph -> ps ) <-> A. x ( ch -> th ) )
5	1	exbii 1658	. . 3 |- ( E. x ph <-> E. x ch )
6	4, 5	anbi12i 464	. 2 |- ( ( A. x ( ph -> ps ) /\ E. x ph ) <-> ( A. x ( ch -> th ) /\ E. x 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. x ( ch -> th ) <-> ( A. x ( ch -> th ) /\ E. x ch ) )
9	6, 7, 8	3bitr4i 212	1 |- ( A.E. x ( ph -> ps ) <-> A.E. x ( 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-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-als 17036

This theorem is referenced by:  2alsraln0m  17066