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Theorem ceqsralv2 30862
 Description: Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.)
Hypothesis
Ref Expression
ceqsralv2.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsralv2 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsralv2
StepHypRef Expression
1 ceqsralv2.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
21notbid 307 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
32ceqsrexv2 30860 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ (𝐴𝐵 ∧ ¬ 𝜓))
4 rexanali 2981 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥𝐵 (𝑥 = 𝐴𝜑))
5 annim 440 . . 3 ((𝐴𝐵 ∧ ¬ 𝜓) ↔ ¬ (𝐴𝐵𝜓))
63, 4, 53bitr3i 289 . 2 (¬ ∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ¬ (𝐴𝐵𝜓))
76con4bii 310 1 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-rex 2902  df-v 3175 This theorem is referenced by: (None)
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