Theorem exbii

Description: Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994)

	Ref	Expression
Hypothesis	exbii.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	exbii	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		exbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2		exbi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 ) )
3	2 1	mpg	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 )