Theorem 19.16 2223

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 2198	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3		albi 1815	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
4	2, 3	syl5bbr 287	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173

This theorem depends on definitions:  df-bi 209  df-ex 1777  df-nf 1781

This theorem is referenced by: (None)