Theorem ralsmd 17138

Description: Deduction rule: Given "all some" applied to a class, the class is inhabited. This is stronger than ralsn0d 17137, which only concludes that the class is nonempty; see n0r 3535. (Contributed by David A. Wheeler, 20-Jul-2026.)

Hypothesis

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

Assertion

Ref	Expression
ralsmd	|- ( ph -> E. x x e. A )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   ch( x)

Proof of Theorem ralsmd

Step	Hyp	Ref	Expression
1		ralsmd.1	. . 3 |- ( ph -> A.E. x e. A ( ps -> ch ) )
2	1	rals2d 17136	. 2 |- ( ph -> E. x e. A ps )
3		rexm 3627	. 2 |- ( E. x e. A ps -> E. x x e. A )
4	2, 3	syl 14	1 |- ( ph -> E. x x e. A )

This proof depends on syntax axioms:   -> wi 4   E.wex 1545   e. wcel 2209   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  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-rex 2534  df-rals 17129

This theorem is used by: (None)