Theorem 19.36

Description: Theorem 19.36 of Margaris p. 90. See 19.36v for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993)

	Ref	Expression
Hypothesis	19.36.1	⊢ Ⅎ 𝑥 𝜓
Assertion	19.36	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.36.1	⊢ Ⅎ 𝑥 𝜓
2		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
3	1	19.9	⊢ ( ∃ 𝑥 𝜓 ↔ 𝜓 )
4	3	imbi2i	⊢ ( ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → 𝜓 ) )
5	2 4	bitri	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → 𝜓 ) )