Theorem bj-hbyfrbi

Description: Version of bj-hbxfrbi with existential quantifiers. (Contributed by BJ, 23-Aug-2023)

	Ref	Expression
Assertion	bj-hbyfrbi	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ) → ( ( ∃ 𝑥 𝜑 → 𝜑 ) ↔ ( ∃ 𝑥 𝜓 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		exbi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 ) )
2	1	adantl	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ) → ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 ) )
3		simpl	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ) → ( 𝜑 ↔ 𝜓 ) )
4	2 3	imbi12d	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ∧ ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) ) → ( ( ∃ 𝑥 𝜑 → 𝜑 ) ↔ ( ∃ 𝑥 𝜓 → 𝜓 ) ) )