Theorem 19.31

Description: Theorem 19.31 of Margaris p. 90. See 19.31v for a version requiring fewer axioms. (Contributed by NM, 14-May-1993)

	Ref	Expression
Hypothesis	19.31.1	⊢ Ⅎ 𝑥 𝜓
Assertion	19.31	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 𝜑 ∨ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.31.1	⊢ Ⅎ 𝑥 𝜓
2	1	19.32	⊢ ( ∀ 𝑥 ( 𝜓 ∨ 𝜑 ) ↔ ( 𝜓 ∨ ∀ 𝑥 𝜑 ) )
3		orcom	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜓 ∨ 𝜑 ) )
4	3	albii	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ∀ 𝑥 ( 𝜓 ∨ 𝜑 ) )
5		orcom	⊢ ( ( ∀ 𝑥 𝜑 ∨ 𝜓 ) ↔ ( 𝜓 ∨ ∀ 𝑥 𝜑 ) )
6	2 4 5	3bitr4i	⊢ ( ∀ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 𝜑 ∨ 𝜓 ) )