Theorem ralsmd 17046

Description: Deduction rule: Given "all some" applied to a class, the class is inhabited. This is stronger than ralsn0d 17045, which only concludes that the class is nonempty; see n0r 3535. (Contributed by David A. Wheeler, 20-Jul-2026.)

Hypothesis

Ref	Expression
ralsmd.1	⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))

Assertion

Ref	Expression
ralsmd	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem ralsmd

Step	Hyp	Ref	Expression
1		ralsmd.1	. . 3 ⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))
2	1	rals2d 17044	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
3		rexm 3627	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 𝑥 ∈ 𝐴)
4	2, 3	syl 14	1 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  ∀∃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  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-rex 2534  df-rals 17037

This theorem is referenced by: (None)