Theorem elv 2624

Description: Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2623), 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 2623	. 2 ⊢ 𝑥 ∈ V
2		elv.1	. 2 ⊢ (𝑥 ∈ V → 𝜑)
3	1, 2	ax-mp 7	1 ⊢ 𝜑

Syntax hints:   → wi 4   ∈ wcel 1439  Vcvv 2620

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-v 2622

This theorem is referenced by:  disjxp1  6015  ixpiinm  6495  ixpsnf1o  6507  iunfidisj  6709  fsum2dlemstep  10882  fsumcnv  10885  fisumcom2  10886  fsumconst  10902  modfsummodlemstep  10905  fsumabs  10913  topnfn  11711  iuncld  11869