Theorem ralrals 17149

Description: If the universal part of a restricted "all some" statement holds, then the statement reduces to the existence of a member of A satisfying its antecedent. This is the restricted counterpart of ralals 17155. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)

Assertion

Ref	Expression
ralrals	|- ( A. x e. A ( ph -> ps ) -> ( A.E. x e. A ( ph -> ps ) <-> E. x e. A ph ) )

Proof of Theorem ralrals

Step	Hyp	Ref	Expression
1		df-rals 17129	. 2 |- ( A.E. x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) )
2		ibar 301	. . 3 |- ( A. x e. A ( ph -> ps ) -> ( E. x e. A ph <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) ) )
3	2	bicomd 141	. 2 |- ( A. x e. A ( ph -> ps ) -> ( ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) <-> E. x e. A ph ) )
4	1, 3	bitrid 192	1 |- ( A. x e. A ( ph -> ps ) -> ( A.E. x e. A ( ph -> ps ) <-> E. x e. A ph ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   A.wral 2528   E.wrex 2529   A.E.wrals 17127

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-rals 17129

This theorem is used by: (None)