Theorem alsbid 17051

Description: Deduction form of alsbii 17049. (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

Ref	Expression
alsbid.1	|- F/ x ph
alsbid.2	|- ( ph -> ( ps <-> th ) )
alsbid.3	|- ( ph -> ( ch <-> ta ) )

Assertion

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

Proof of Theorem alsbid

Step	Hyp	Ref	Expression
1		alsbid.1	. . . 4 |- F/ x ph
2		alsbid.2	. . . . 5 |- ( ph -> ( ps <-> th ) )
3		alsbid.3	. . . . 5 |- ( ph -> ( ch <-> ta ) )
4	2, 3	imbi12d 234	. . . 4 |- ( ph -> ( ( ps -> ch ) <-> ( th -> ta ) ) )
5	1, 4	albid 1668	. . 3 |- ( ph -> ( A. x ( ps -> ch ) <-> A. x ( th -> ta ) ) )
6	1, 2	exbid 1669	. . 3 |- ( ph -> ( E. x ps <-> E. x th ) )
7	5, 6	anbi12d 477	. 2 |- ( ph -> ( ( A. x ( ps -> ch ) /\ E. x ps ) <-> ( A. x ( th -> ta ) /\ E. x th ) ) )
8		df-als 17036	. 2 |- ( A.E. x ( ps -> ch ) <-> ( A. x ( ps -> ch ) /\ E. x ps ) )
9		df-als 17036	. 2 |- ( A.E. x ( th -> ta ) <-> ( A. x ( th -> ta ) /\ E. x th ) )
10	7, 8, 9	3bitr4g 223	1 |- ( ph -> ( A.E. x ( ps -> ch ) <-> A.E. x ( th -> ta ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   F/wnf 1513   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-nf 1514  df-als 17036

This theorem is referenced by: (None)