Theorem 19.17 1544

Description: Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Hypothesis

Ref	Expression
19.17.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
19.17	⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓))

Proof of Theorem 19.17

Step	Hyp	Ref	Expression
1		albi 1456	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
2		19.17.1	. . 3 ⊢ Ⅎ𝑥𝜓
3	2	19.3 1542	. 2 ⊢ (∀𝑥𝜓 ↔ 𝜓)
4	1, 3	bitrdi 195	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  Ⅎwnf 1448

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 1435  ax-gen 1437  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by: (None)