Theorem rals1d 17135

Description: Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)

Hypothesis

Ref	Expression
rals1d.1	|- ( ph -> A.E. x e. A ( ps -> ch ) )

Assertion

Ref	Expression
rals1d	|- ( ph -> A. x e. A ( ps -> ch ) )

Proof of Theorem rals1d

Step	Hyp	Ref	Expression
1		rals1d.1	. . 3 |- ( ph -> A.E. x e. A ( ps -> ch ) )
2		df-rals 17129	. . 3 |- ( A.E. x e. A ( ps -> ch ) <-> ( A. x e. A ( ps -> ch ) /\ E. x e. A ps ) )
3	1, 2	sylib 122	. 2 |- ( ph -> ( A. x e. A ( ps -> ch ) /\ E. x e. A ps ) )
4	3	simpld 112	1 |- ( ph -> A. x e. A ( ps -> ch ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   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

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

This theorem is used by: (None)