Theorem rspcdv 3452

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 219	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
4	1, 3	rspcimdv 3450	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342

This theorem is referenced by:  ralxfrd  5028  ralxfrdOLD  5029  ralxfrd2  5033  suppofss1d  7501  suppofss2d  7502  zindd  11670  wrd2ind  13677  ismri2dad  16499  mreexd  16504  mreexexlemd  16506  catcocl  16547  catass  16548  moni  16597  subccocl  16706  funcco  16732  fullfo  16773  fthf1  16778  nati  16816  mrcmndind  17567  idsrngd  19064  flfcntr  22048  uspgr2wlkeq  26752  crctcshwlkn0lem4  26916  crctcshwlkn0lem5  26917  wwlknllvtx  26950  0enwwlksnge1  26973  wlkiswwlks2lem5  26982  clwlkclwwlklem2a  27121  clwlkclwwlklem2  27123  clwwisshclwws  27138  clwwlkinwwlk  27169  umgr2cwwk2dif  27195  rngurd  30097  esumcvg  30457  inelcarsg  30682  carsgclctunlem1  30688  orvcelel  30840  signsply0  30937  onint1  32754  rspcdvinvd  38976  ralbinrald  41705  fargshiftfva  41889  evengpop3  42196  evengpoap3  42197  snlindsntorlem  42769