Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ceqsralv Structured version   Visualization version   GIF version

Theorem ceqsralv 3220
 Description: Restricted quantifier version of ceqsalv 3219. (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 1840 . 2 𝑥𝜓
2 ceqsralv.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32ax-gen 1719 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 ceqsralt 3215 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝐵) → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
51, 3, 4mp3an12 1411 1 (𝐴𝐵 → (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478   = wceq 1480  Ⅎwnf 1705   ∈ wcel 1987  ∀wral 2907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-12 2044  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-v 3188 This theorem is referenced by:  eqreu  3380  sqrt2irr  14903  acsfn  16241  ovolgelb  23155
 Copyright terms: Public domain W3C validator