Theorem 19.31 2089

Description: Theorem 19.31 of [Margaris] p. 90. See 19.31v 1857 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.)

Hypothesis

Ref	Expression
19.31.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
19.31	⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))

Proof of Theorem 19.31

Step	Hyp	Ref	Expression
1		19.31.1	. . 3 ⊢ Ⅎ𝑥𝜓
2	1	19.32 2088	. 2 ⊢ (∀𝑥(𝜓 ∨ 𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑))
3		orcom 401	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
4	3	albii 1737	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(𝜓 ∨ 𝜑))
5		orcom 401	. 2 ⊢ ((∀𝑥𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
6	2, 4, 5	3bitr4i 291	1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))

Syntax hints:   ↔ wb 195   ∨ wo 382  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701

This theorem is referenced by:  2eu3  2543