Theorem rexrals 17150

Description: If a member of A satisfying the antecedent exists, then a restricted "all some" statement reduces to its universal part. This is the restricted counterpart of rexals 17156. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)

Assertion

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

Proof of Theorem rexrals

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		iba 300	. . 3 |- ( E. x e. A ph -> ( A. x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) ) )
3	2	bicomd 141	. 2 |- ( E. x e. A ph -> ( ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) <-> A. x e. A ( ph -> ps ) ) )
4	1, 3	bitrid 192	1 |- ( E. x e. A ph -> ( A.E. x e. A ( ph -> ps ) <-> A. x e. A ( ph -> ps ) ) )

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)