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Theorem ceqsralv2 31733
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 31731 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ (𝐴𝐵 ∧ ¬ 𝜓))
4 rexanali 3027 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥𝐵 (𝑥 = 𝐴𝜑))
5 annim 440 . . 3 ((𝐴𝐵 ∧ ¬ 𝜓) ↔ ¬ (𝐴𝐵𝜓))
63, 4, 53bitr3i 290 . 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-ral 2946  df-rex 2947  df-v 3233
This theorem is referenced by: (None)
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