Theorem 19.45

Description: Theorem 19.45 of Margaris p. 90. See 19.45v for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Hypothesis	19.45.1	⊢ Ⅎ 𝑥 𝜑
Assertion	19.45	⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.45.1	⊢ Ⅎ 𝑥 𝜑
2		19.43	⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) )
3	1	19.9	⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 )
4	3	orbi1i	⊢ ( ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) ↔ ( 𝜑 ∨ ∃ 𝑥 𝜓 ) )
5	2 4	bitri	⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∃ 𝑥 𝜓 ) )