Theorem 19.45v

Description: Version of 19.45 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Assertion	19.45v	⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ∃ 𝑥 𝜓 ) )

Proof

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