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Theorem ralrexbidOLD 3282
 Description: Obsolete version of ralrexbid 3281 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 → (𝜓𝜃))
Assertion
Ref Expression
ralrexbidOLD (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))

Proof of Theorem ralrexbidOLD
StepHypRef Expression
1 nfra1 3183 . 2 𝑥𝑥𝐴 𝜑
2 rspa 3171 . . 3 ((∀𝑥𝐴 𝜑𝑥𝐴) → 𝜑)
3 ralrexbid.1 . . 3 (𝜑 → (𝜓𝜃))
42, 3syl 17 . 2 ((∀𝑥𝐴 𝜑𝑥𝐴) → (𝜓𝜃))
51, 4rexbida 3277 1 (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-ral 3111  df-rex 3112 This theorem is referenced by: (None)
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