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Theorem ceqsralv 3459
Description: Restricted quantifier version of ceqsalv 3457. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2118, ax-12 2173, ax-ext 2709. (Revised by SN, 8-Sep-2024.)
Hypothesis
Ref Expression
ceqsralv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsralv (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsralv
StepHypRef Expression
1 ceqsralv.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
21pm5.74i 270 . . 3 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
32ralbii 3090 . 2 (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ ∀𝑥𝐵 (𝑥 = 𝐴𝜓))
4 r19.23v 3207 . . 3 (∀𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ (∃𝑥𝐵 𝑥 = 𝐴𝜓))
5 risset 3193 . . . 4 (𝐴𝐵 ↔ ∃𝑥𝐵 𝑥 = 𝐴)
6 pm5.5 361 . . . 4 (∃𝑥𝐵 𝑥 = 𝐴 → ((∃𝑥𝐵 𝑥 = 𝐴𝜓) ↔ 𝜓))
75, 6sylbi 216 . . 3 (𝐴𝐵 → ((∃𝑥𝐵 𝑥 = 𝐴𝜓) ↔ 𝜓))
84, 7syl5bb 282 . 2 (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜓) ↔ 𝜓))
93, 8syl5bb 282 1 (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2108  wral 3063  wrex 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-clel 2817  df-ral 3068  df-rex 3069
This theorem is referenced by:  eqreu  3659  sqrt2irr  15886  acsfn  17285  ovolgelb  24549  fsuppind  40202
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