Theorem exbidh

Description: Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 26-May-1993)

	Ref	Expression
Hypotheses	exbidh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	exbidh.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	exbidh	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) )

Proof

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