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Theorem ceqsralv 3229
Description: Restricted quantifier version of ceqsalv 3228. (Contributed by NM, 21-Jun-2013.)
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 nfv 1841 . 2 𝑥𝜓
2 ceqsralv.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32ax-gen 1720 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 ceqsralt 3224 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 3, 4mp3an12 1412 1 (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479   = wceq 1481  wnf 1706  wcel 1988  wral 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-12 2045  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-ral 2914  df-v 3197
This theorem is referenced by:  eqreu  3392  sqrt2irr  14960  acsfn  16301  ovolgelb  23229
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