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Theorem ceqsralv2 31914
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 31912 . . 3 (∃𝑥𝐵 (𝑥 = 𝐴 ∧ ¬ 𝜑) ↔ (𝐴𝐵 ∧ ¬ 𝜓))
4 rexanali 3136 . . 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 1632  wcel 2139  wral 3050  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-rex 3056  df-v 3342
This theorem is referenced by: (None)
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