Theorem elv 2690

Description: Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2689), 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 2689	. 2 ⊢ 𝑥 ∈ V
2		elv.1	. 2 ⊢ (𝑥 ∈ V → 𝜑)
3	1, 2	ax-mp 5	1 ⊢ 𝜑

Syntax hints:   → wi 4   ∈ wcel 1480  Vcvv 2686

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688

This theorem is referenced by:  xpiindim  4676  disjxp1  6133  ixpiinm  6618  ixpsnf1o  6630  iunfidisj  6834  ssfii  6862  fifo  6868  omp1eomlem  6979  exmidomniim  7013  bcval5  10509  rexfiuz  10761  fsum2dlemstep  11203  fsumcnv  11206  fisumcom2  11207  fsumconst  11223  modfsummodlemstep  11226  fsumabs  11234  ennnfonelemim  11937  topnfn  12125  iuncld  12284  txbas  12427  txdis  12446  xmetunirn  12527  xmettxlem  12678  xmettx  12679