Theorem ralsbii 17050

Description: Congruence for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

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

Assertion

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

Proof of Theorem ralsbii

Step	Hyp	Ref	Expression
1		ralsbii.1	. . . . 5 |- ( ph <-> ch )
2		ralsbii.2	. . . . 5 |- ( ps <-> th )
3	1, 2	imbi12i 239	. . . 4 |- ( ( ph -> ps ) <-> ( ch -> th ) )
4	3	ralbii 2556	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x e. A ( ch -> th ) )
5	1	rexbii 2557	. . 3 |- ( E. x e. A ph <-> E. x e. A ch )
6	4, 5	anbi12i 464	. 2 |- ( ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) <-> ( A. x e. A ( ch -> th ) /\ E. x e. A ch ) )
7		df-rals 17037	. 2 |- ( A.E. x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) )
8		df-rals 17037	. 2 |- ( A.E. x e. A ( ch -> th ) <-> ( A. x e. A ( ch -> th ) /\ E. x e. A ch ) )
9	6, 7, 8	3bitr4i 212	1 |- ( A.E. x e. A ( ph -> ps ) <-> A.E. x e. A ( ch -> th ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wral 2528   E.wrex 2529   A.E.wrals 17035

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-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-ral 2533  df-rex 2534  df-rals 17037

This theorem is referenced by: (None)