Theorem ra4v 3836

Description: Version of ra4 3837 with a disjoint variable condition, requiring fewer axioms. This is stdpc5v 1973 for a restricted domain. (Contributed by BJ, 27-Mar-2020.)

Assertion

Ref	Expression
ra4v	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem ra4v

Step	Hyp	Ref	Expression
1		r19.21v 3191	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
2	1	biimpi 219	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wral 3080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3081

This theorem is used by:  wfr3g  8375  frr3g  9797  eqhf  35861  eqhfregs  35946