ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elv GIF version

Theorem elv 2741
Description: Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2740), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.)
Hypothesis
Ref Expression
elv.1 (𝑥 ∈ V → 𝜑)
Assertion
Ref Expression
elv 𝜑

Proof of Theorem elv
StepHypRef Expression
1 vex 2740 . 2 𝑥 ∈ V
2 elv.1 . 2 (𝑥 ∈ V → 𝜑)
31, 2ax-mp 5 1 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  Vcvv 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739
This theorem is referenced by:  xpiindim  4764  disjxp1  6236  ixpiinm  6723  ixpsnf1o  6735  iunfidisj  6944  ssfii  6972  fifo  6978  dcfi  6979  omp1eomlem  7092  exmidomniim  7138  bcval5  10742  rexfiuz  10997  fsum2dlemstep  11441  fsumcnv  11444  fisumcom2  11445  fsumconst  11461  modfsummodlemstep  11464  fsumabs  11472  fprodcllemf  11620  fprod2dlemstep  11629  fprodcnv  11632  fprodcom2fi  11633  fprodmodd  11648  ennnfonelemim  12424  topnfn  12692  ptex  12712  xpsff1o  12767  ismgm  12775  issgrp  12808  ismnddef  12818  isnsg  13060  fnmgp  13130  isring  13181  iuncld  13585  txbas  13728  txdis  13747  xmetunirn  13828  xmettxlem  13979  xmettx  13980  pw1nct  14722
  Copyright terms: Public domain W3C validator