Theorem 19.32v

Description: Version of 19.32 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020)

	Ref	Expression
Assertion	19.32v	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.21v	⊢ ( ∀ 𝑥 ( ¬ 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 → ∀ 𝑥 𝜓 ) )
2		df-or	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 → 𝜓 ) )
3	2	albii	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ∀ 𝑥 ( ¬ 𝜑 → 𝜓 ) )
4		df-or	⊢ ( ( 𝜑 ∨ ∀ 𝑥 𝜓 ) ↔ ( ¬ 𝜑 → ∀ 𝑥 𝜓 ) )
5	1 3 4	3bitr4i	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∀ 𝑥 𝜓 ) )