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

Theorem ceqsexg 3597
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
Hypotheses
Ref Expression
ceqsexg.1 𝑥𝜓
ceqsexg.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsexg (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsexg
StepHypRef Expression
1 nfe1 2152 . . 3 𝑥𝑥(𝑥 = 𝐴𝜑)
2 ceqsexg.1 . . 3 𝑥𝜓
31, 2nfbi 1904 . 2 𝑥(∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
4 ceqex 3596 . . 3 (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
5 ceqsexg.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
64, 5bibi12d 349 . 2 (𝑥 = 𝐴 → ((𝜑𝜑) ↔ (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)))
7 biid 264 . 2 (𝜑𝜑)
83, 6, 7vtoclg1f 3517 1 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wnf 1785  wcel 2112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446
This theorem is referenced by:  ceqsexgvOLD  3599
  Copyright terms: Public domain W3C validator