Theorem 19.37v

Description: Version of 19.37 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Assertion	19.37v	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
2		19.3v	⊢ ( ∀ 𝑥 𝜑 ↔ 𝜑 )
3	2	imbi1i	⊢ ( ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) ↔ ( 𝜑 → ∃ 𝑥 𝜓 ) )
4	1 3	bitri	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∃ 𝑥 𝜓 ) )