Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  el3v Structured version   Visualization version   GIF version

Theorem el3v 35609
Description: New way (elv 3474, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. Inference forms (with 𝐴 ∈ V, 𝐵 ∈ V and 𝐶 ∈ V hypotheses) of the general theorems (proving ((𝐴𝑉𝐵𝑊𝐶𝑋) → assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.)
Hypothesis
Ref Expression
el3v.1 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)
Assertion
Ref Expression
el3v 𝜑

Proof of Theorem el3v
StepHypRef Expression
1 vex 3472 . 2 𝑥 ∈ V
2 vex 3472 . 2 𝑦 ∈ V
3 vex 3472 . 2 𝑧 ∈ V
4 el3v.1 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)
51, 2, 3, 4mp3an 1458 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2114  Vcvv 3469
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 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471
This theorem is referenced by:  dfxrn2  35746
  Copyright terms: Public domain W3C validator