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 34333
Description: New way (elv 34329, 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 3354 . 2 𝑥 ∈ V
2 vex 3354 . 2 𝑦 ∈ V
3 vex 3354 . 2 𝑧 ∈ V
4 el3v.1 . 2 ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)
51, 2, 3, 4mp3an 1572 1 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071  wcel 2145  Vcvv 3351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-3an 1073  df-tru 1634  df-ex 1853  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353
This theorem is referenced by:  dfxrn2  34481
  Copyright terms: Public domain W3C validator