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

Theorem bj-vtoclf 34128
Description: Remove dependency on ax-ext 2790, df-clab 2797 and df-cleq 2811 (and df-sb 2061 and df-v 3494) from vtoclf 3556. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-vtoclf.nf 𝑥𝜓
bj-vtoclf.s 𝐴𝑉
bj-vtoclf.maj (𝑥 = 𝐴 → (𝜑𝜓))
bj-vtoclf.min 𝜑
Assertion
Ref Expression
bj-vtoclf 𝜓
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-vtoclf
StepHypRef Expression
1 bj-vtoclf.nf . . 3 𝑥𝜓
2 bj-vtoclf.s . . . . 5 𝐴𝑉
32bj-issetiv 34090 . . . 4 𝑥 𝑥 = 𝐴
4 bj-vtoclf.maj . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
54biimpd 230 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
63, 5eximii 1828 . . 3 𝑥(𝜑𝜓)
71, 619.36i 2223 . 2 (∀𝑥𝜑𝜓)
8 bj-vtoclf.min . 2 𝜑
97, 8mpg 1789 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wnf 1775  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-clel 2890
This theorem is referenced by:  bj-vtocl  34129
  Copyright terms: Public domain W3C validator