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Theorem ceqsralbv 3645
Description: Elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.)
Hypothesis
Ref Expression
ceqsrexv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsralbv (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsralbv
StepHypRef Expression
1 ceqsrexv.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
21notbid 317 . . . 4 (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
32ceqsrexbv 3644 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ (𝐴𝐵 ∧ ¬ 𝜓))
4 rexanali 3102 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥𝐵 (𝑥 = 𝐴𝜑))
5 annim 404 . . 3 ((𝐴𝐵 ∧ ¬ 𝜓) ↔ ¬ (𝐴𝐵𝜓))
63, 4, 53bitr3i 300 . 2 (¬ ∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ¬ (𝐴𝐵𝜓))
76con4bii 320 1 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071
This theorem is referenced by:  ref5  37177
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