Theorem 19.31 2235

Description: Theorem 19.31 of [Margaris] p. 90. See 19.31v 1941 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 2234	. 2 ⊢ (∀𝑥(𝜓 ∨ 𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑))
3		orcom 870	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
4	3	albii 1819	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(𝜓 ∨ 𝜑))
5		orcom 870	. 2 ⊢ ((∀𝑥𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
6	2, 4, 5	3bitr4i 303	1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))

Syntax hints:   ↔ wb 206   ∨ wo 847  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1780  df-nf 1784

This theorem is referenced by:  2eu3  2654