Theorem 19.31v

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

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

Proof

Step	Hyp	Ref	Expression
1		19.32v	⊢ ( ∀ 𝑥 ( 𝜓 ∨ 𝜑 ) ↔ ( 𝜓 ∨ ∀ 𝑥 𝜑 ) )
2		orcom	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜓 ∨ 𝜑 ) )
3	2	albii	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ∀ 𝑥 ( 𝜓 ∨ 𝜑 ) )
4		orcom	⊢ ( ( ∀ 𝑥 𝜑 ∨ 𝜓 ) ↔ ( 𝜓 ∨ ∀ 𝑥 𝜑 ) )
5	1 3 4	3bitr4i	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 𝜑 ∨ 𝜓 ) )