Theorem r19.21 3259

Description: Restricted quantifier version of 19.21 2244. (Contributed by Scott Fenton, 30-Mar-2011.)

Hypothesis

Ref	Expression
r19.21.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
r19.21	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Proof of Theorem r19.21

Step	Hyp	Ref	Expression
1		r19.21.1	. 2 ⊢ Ⅎ𝑥𝜑
2		r19.21t 3258	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 208  Ⅎwnf 1805  ∀wral 3078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-12 2214

This theorem depends on definitions:  df-bi 209  df-ex 1802  df-nf 1806  df-ral 3079

This theorem is referenced by:  rmo3f  3699  ra4  3841  rmoanim  3849  rmoanimALT  3850  r19.32  47697