Theorem 19.17 2094

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 1746	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
2		19.17.1	. . 3 ⊢ Ⅎ𝑥𝜓
3	2	19.3 2069	. 2 ⊢ (∀𝑥𝜓 ↔ 𝜓)
4	1, 3	syl6bb 276	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)