Theorem dfrals2 17038

Description: The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)

Assertion

Ref	Expression
dfrals2	|- ( A.E. x e. A ( ph -> ps ) <-> A.E. x ( ( x e. A /\ ph ) -> ps ) )

Proof of Theorem dfrals2

Step	Hyp	Ref	Expression
1		df-ral 2533	. . . 4 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		impexp 263	. . . . 5 |- ( ( ( x e. A /\ ph ) -> ps ) <-> ( x e. A -> ( ph -> ps ) ) )
3	2	albii 1523	. . . 4 |- ( A. x ( ( x e. A /\ ph ) -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
4	1, 3	bitr4i 187	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ps ) )
5		df-rex 2534	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6	4, 5	anbi12i 464	. 2 |- ( ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) <-> ( A. x ( ( x e. A /\ ph ) -> ps ) /\ E. x ( x e. A /\ ph ) ) )
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-als 17036	. 2 |- ( A.E. x ( ( x e. A /\ ph ) -> ps ) <-> ( A. x ( ( x e. A /\ ph ) -> ps ) /\ E. x ( x e. A /\ ph ) ) )
9	6, 7, 8	3bitr4i 212	1 |- ( A.E. x e. A ( ph -> ps ) <-> A.E. x ( ( x e. A /\ ph ) -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   A.E.wals 17034   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

This theorem depends on definitions:  df-bi 117  df-ral 2533  df-rex 2534  df-als 17036  df-rals 17037

This theorem is referenced by:  rals-no-surprise  17056