Theorem 19.31 2227

Description: Theorem 19.31 of [Margaris] p. 90. See 19.31v 1944 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 2226	. 2 ⊢ (∀𝑥(𝜓 ∨ 𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑))
3		orcom 867	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
4	3	albii 1822	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(𝜓 ∨ 𝜑))
5		orcom 867	. 2 ⊢ ((∀𝑥𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
6	2, 4, 5	3bitr4i 303	1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))

Syntax hints:   ↔ wb 205   ∨ wo 844  ∀wal 1537  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by:  2eu3  2655