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Theorem ralrexbidOLD 3105
Description: Obsolete version of ralrexbid 3104 as of 4-Nov-2024. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-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 df-ral 3060 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 ralrexbid.1 . . . . . 6 (𝜑 → (𝜓𝜃))
32imim2i 16 . . . . 5 ((𝑥𝐴𝜑) → (𝑥𝐴 → (𝜓𝜃)))
43pm5.32d 577 . . . 4 ((𝑥𝐴𝜑) → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜃)))
54alexbii 1830 . . 3 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜃)))
61, 5sylbi 217 . 2 (∀𝑥𝐴 𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑥(𝑥𝐴𝜃)))
7 df-rex 3069 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
8 df-rex 3069 . 2 (∃𝑥𝐴 𝜃 ↔ ∃𝑥(𝑥𝐴𝜃))
96, 7, 83bitr4g 314 1 (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wex 1776  wcel 2106  wral 3059  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-ral 3060  df-rex 3069
This theorem is referenced by: (None)
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