Theorem 19.31v 1944

Description: Version of 19.31 2227 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)

Assertion

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

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.31v

Step	Hyp	Ref	Expression
1		19.32v 1943	. 2 ⊢ (∀𝑥(𝜓 ∨ 𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑))
2		orcom 867	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
3	2	albii 1822	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(𝜓 ∨ 𝜑))
4		orcom 867	. 2 ⊢ ((∀𝑥𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
5	1, 3, 4	3bitr4i 303	1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))

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

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

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

This theorem is referenced by:  19.31vv  42002