Theorem rspcdv 3605

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
rspcdv.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspcdv.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rspcdv	⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcdv

Step	Hyp	Ref	Expression
1		rspcdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		rspcdv.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3	2	biimpd 228	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
4	1, 3	rspcimdv 3603	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063

This theorem is referenced by:  rspcdv2  3608  rspcv  3609  ralxfrd  5407  ralxfrd2  5411  reuop  6293  suppofss1d  8189  suppofss2d  8190  zindd  12663  wrd2ind  14673  ismri2dad  17581  mreexd  17586  mreexexlemd  17588  catcocl  17629  catass  17630  moni  17683  subccocl  17795  funcco  17821  fullfo  17863  fthf1  17868  nati  17906  mndind  18709  ringurd  20008  idsrngd  20470  uspgr2wlkeq  28903  crctcshwlkn0lem4  29067  crctcshwlkn0lem5  29068  wwlknllvtx  29100  0enwwlksnge1  29118  wlkiswwlks2lem5  29127  clwlkclwwlklem2a  29251  clwlkclwwlklem2  29253  clwwisshclwws  29268  clwwlkinwwlk  29293  umgr2cwwk2dif  29317  wrdt2ind  32117  mgccole1  32160  mgccole2  32161  mgcmnt1  32162  mgcmntco  32164  dfmgc2lem  32165  fedgmullem2  32715  zart0  32859  zarcmplem  32861  esumcvg  33084  inelcarsg  33310  carsgclctunlem1  33316  orvcelel  33468  signsply0  33562  mpomulcn  35162  onint1  35334  qsalrel  41062  ismnushort  43060  ralbinrald  45830  fargshiftfva  46111  reupr  46190  evengpop3  46466  evengpoap3  46467  isomgrsym  46504  snlindsntorlem  47151