Theorem ralrexbidOLD 3322

Description: Obsolete version of ralrexbid 3321 as of 13-Nov-2023. (Contributed by AV, 21-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ralrexbid.1	⊢ (𝜑 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
ralrexbidOLD	⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))

Proof of Theorem ralrexbidOLD

Step	Hyp	Ref	Expression
1		nfra1 3218	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
2		rspa 3205	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜑)
3		ralrexbid.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜃))
4	2, 3	syl 17	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜃))
5	1, 4	rexbida 3317	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2113  ∀wral 3137  ∃wrex 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-ral 3142  df-rex 3143

This theorem is referenced by: (None)