Theorem r19.21 2437

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (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 2436	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)))
3	1, 2	ax-mp 7	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 103  Ⅎwnf 1389  ∀wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353

This theorem is referenced by:  r19.21v  2438