Theorem 19.16 1534

Description: Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Hypothesis

Ref	Expression
19.16.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.16	⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓))

Proof of Theorem 19.16

Step	Hyp	Ref	Expression
1		19.16.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	19.3 1533	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3		albi 1444	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4	2, 3	syl5bbr 193	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by: (None)