Theorem 19.17 1489

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 1398	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
2		19.17.1	. . 3 ⊢ Ⅎ𝑥𝜓
3	2	19.3 1487	. 2 ⊢ (∀𝑥𝜓 ↔ 𝜓)
4	1, 3	syl6bb 194	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1283  Ⅎwnf 1390

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 1377  ax-gen 1379  ax-4 1441

This theorem depends on definitions:  df-bi 115  df-nf 1391

This theorem is referenced by: (None)