Theorem alexbii

Description: Biconditional form of aleximi . (Contributed by BJ, 16-Nov-2020)

	Ref	Expression
Hypothesis	alexbii.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	alexbii	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		alexbii.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	biimpd	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3	2	aleximi	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )
4	1	biimprd	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
5	4	aleximi	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜒 → ∃ 𝑥 𝜓 ) )
6	3 5	impbid	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) )