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Theorem ceqsralv2 30665
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 306 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
32ceqsrexv2 30663 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ (𝐴𝐵 ∧ ¬ 𝜓))
4 rexanali 2977 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥𝐵 (𝑥 = 𝐴𝜑))
5 annim 439 . . 3 ((𝐴𝐵 ∧ ¬ 𝜓) ↔ ¬ (𝐴𝐵𝜓))
63, 4, 53bitr3i 288 . 2 (¬ ∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ¬ (𝐴𝐵𝜓))
76con4bii 309 1 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2892  wrex 2893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2032  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-ral 2897  df-rex 2898  df-v 3171
This theorem is referenced by: (None)
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