Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-vtoclg1f Structured version   Visualization version   GIF version

Theorem bj-vtoclg1f 34359
 Description: Reprove vtoclg1f 3517 from bj-vtoclg1f1 34358. This removes dependency on ax-ext 2773, df-cleq 2794 and df-v 3446. Use bj-vtoclg1fv 34360 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-vtoclg1f.nf 𝑥𝜓
bj-vtoclg1f.maj (𝑥 = 𝐴 → (𝜑𝜓))
bj-vtoclg1f.min 𝜑
Assertion
Ref Expression
bj-vtoclg1f (𝐴𝑉𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem bj-vtoclg1f
StepHypRef Expression
1 bj-elisset 34317 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 bj-vtoclg1f.nf . . 3 𝑥𝜓
3 bj-vtoclg1f.maj . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 bj-vtoclg1f.min . . 3 𝜑
52, 3, 4bj-exlimmpi 34353 . 2 (∃𝑥 𝑥 = 𝐴𝜓)
61, 5syl 17 1 (𝐴𝑉𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ∃wex 1781  Ⅎwnf 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-12 2176 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-clel 2873 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator