Theorem exbid

Description: Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016)

	Ref	Expression
Hypotheses	albid.1	⊢ Ⅎ 𝑥 𝜑
	albid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	exbid	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		albid.1	⊢ Ⅎ 𝑥 𝜑
2		albid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	1	nf5ri	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
4	3 2	exbidh	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) )