Theorem ralrexbidOLD

Description: Obsolete version of ralrexbid as of 13-Nov-2023. (Contributed by AV, 21-Oct-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	ralrexbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
Assertion	ralrexbidOLD	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		ralrexbid.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜃 ) )
2		nfra1	⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 𝜑
3		rspa	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜑 )
4	3 1	syl	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜃 ) )
5	2 4	rexbida	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜃 ) )