Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  spcgv GIF version

Theorem spcgv 2799
 Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
Hypothesis
Ref Expression
spcgv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcgv (𝐴𝑉 → (∀𝑥𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem spcgv
StepHypRef Expression
1 nfcv 2299 . 2 𝑥𝐴
2 nfv 1508 . 2 𝑥𝜓
3 spcgv.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3spcgf 2794 1 (𝐴𝑉 → (∀𝑥𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333   = wceq 1335   ∈ wcel 2128 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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714 This theorem is referenced by:  spcv  2806  mob2  2892  intss1  3822  dfiin2g  3882  exmidsssnc  4163  frirrg  4309  frind  4311  alxfr  4419  elirr  4498  en2lp  4511  tfisi  4544  mptfvex  5550  tfrcl  6305  rdgisucinc  6326  frecabex  6339  fisseneq  6869  mkvprop  7084  exmidfodomrlemr  7120  exmidfodomrlemrALT  7121  acfun  7125  ccfunen  7167  zfz1isolem1  10693  zfz1iso  10694  uniopn  12359  exmid1stab  13533  pw1nct  13536  sbthom  13560
 Copyright terms: Public domain W3C validator