Theorem r19.21 2956

Description: Restricted quantifier version of 19.21 2075. (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 2955	. 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 196  Ⅎwnf 1708  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710  df-ral 2917

This theorem is referenced by:  ra4  3525  rmo3f  29335  rmo4fOLD  29336  r19.32  41167  rmoanim  41179