Theorem elv 2734

Description: Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2733), 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

Step	Hyp	Ref	Expression
1		vex 2733	. 2 ⊢ 𝑥 ∈ V
2		elv.1	. 2 ⊢ (𝑥 ∈ V → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  xpiindim  4748  disjxp1  6215  ixpiinm  6702  ixpsnf1o  6714  iunfidisj  6923  ssfii  6951  fifo  6957  dcfi  6958  omp1eomlem  7071  exmidomniim  7117  bcval5  10697  rexfiuz  10953  fsum2dlemstep  11397  fsumcnv  11400  fisumcom2  11401  fsumconst  11417  modfsummodlemstep  11420  fsumabs  11428  fprodcllemf  11576  fprod2dlemstep  11585  fprodcnv  11588  fprodcom2fi  11589  fprodmodd  11604  ennnfonelemim  12379  topnfn  12584  ismgm  12611  issgrp  12644  ismnddef  12654  iuncld  12909  txbas  13052  txdis  13071  xmetunirn  13152  xmettxlem  13303  xmettx  13304  pw1nct  14036