Theorem ra4 3869

Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is the axiom stdpc5 2208 of standard predicate calculus for a restricted domain. See ra4v 3868 for a version requiring fewer axioms. (Contributed by NM, 16-Jan-2004.) (Proof shortened by BJ, 27-Mar-2020.)

Hypothesis

Ref	Expression
ra4.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
ra4	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Proof of Theorem ra4

Step	Hyp	Ref	Expression
1		ra4.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	r19.21 3215	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
3	2	biimpi 218	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4  Ⅎwnf 1784  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785  df-ral 3143

This theorem is referenced by: (None)