Theorem 19.37

Description: Theorem 19.37 of Margaris p. 90. See 19.37v for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypothesis	19.37.1	⊢ Ⅎ 𝑥 𝜑
Assertion	19.37	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.37.1	⊢ Ⅎ 𝑥 𝜑
2		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
3	1	19.3	⊢ ( ∀ 𝑥 𝜑 ↔ 𝜑 )
4	3	imbi1i	⊢ ( ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) ↔ ( 𝜑 → ∃ 𝑥 𝜓 ) )
5	2 4	bitri	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∃ 𝑥 𝜓 ) )