Theorem elvd 3459

Description: If a proposition is implied by 𝑥 ∈ V (which is true, see vex 3457) and another antecedent, then it is implied by that other antecedent. Deduction associated with elv 3458. (Contributed by Peter Mazsa, 23-Oct-2018.)

Hypothesis

Ref	Expression
elvd.1	⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓)

Assertion

Ref	Expression
elvd	⊢ (𝜑 → 𝜓)

Proof of Theorem elvd

Step	Hyp	Ref	Expression
1		vex 3457	. 2 ⊢ 𝑥 ∈ V
2		elvd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓)
3	1, 2	mpan2 701	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  Vcvv 3453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455

This theorem is referenced by:  inimasn  6138  predep  6313  dffv3  6859  dmfco  6959  fsnex  7263  2ndconst  8075  curry1  8078  qsel  8773  ralxpmap  8874  domunsn  9095  dif1ennnALT  9217  eqinf  9428  dfacacn  10095  dfac13  10096  intgru  10769  shftfib  15082  rlimdm  15561  mat1scmat  22579  imasnopn  23730  imasncld  23731  imasncls  23732  ustuqtop1  24281  ustuqtop2  24282  ustuqtop3  24283  blval2  24602  mulsval  28179  dfnbgr2  29484  nbuhgr  29490  iunsnima2  32771  gblacfnacd  35409  vonf1wev  35415  vonf1owevOLD  35417  vonf1oonfo  35422  fmlasucdisj  35713  opelco3  36089  funpartfv  36259  tailfb  36701  el3v23  38697  eldm4  38744  eldmcnv  38808  ecin0  38815  ecun  38856  ecxrn2  38871  ecqmap  38912  dfpre2  38940  brcoss3  38986  refressn  38996  disjlem19  39367  petseq  39439  pwslnmlem1  43633  rlimdmafv  47735  dfatsnafv2  47810  dfafv23  47811  dfatdmfcoafv2  47812  rlimdmafv2  47816  dfclnbgr2  48409  uspgrsprfo  48734