Theorem 19.19

Description: Theorem 19.19 of Margaris p. 90. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Hypothesis	19.19.1	⊢ Ⅎ 𝑥 𝜑
Assertion	19.19	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ↔ ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.19.1	⊢ Ⅎ 𝑥 𝜑
2	1	19.9	⊢ ( ∃ 𝑥 𝜑 ↔ 𝜑 )
3		exbi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 ) )
4	2 3	syl5bbr	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ↔ ∃ 𝑥 𝜓 ) )