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

Theorem spcgv 2634
 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 (A 𝑉 → (xφψ))
Distinct variable groups:   ψ,x   x,A
Allowed substitution hints:   φ(x)   𝑉(x)

Proof of Theorem spcgv
StepHypRef Expression
1 nfcv 2175 . 2 xA
2 nfv 1418 . 2 xψ
3 spcgv.1 . 2 (x = A → (φψ))
41, 2, 3spcgf 2629 1 (A 𝑉 → (xφψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553 This theorem is referenced by:  spcv  2640  mob2  2715  intss1  3621  dfiin2g  3681  alxfr  4159  elirr  4224  en2lp  4232  tfisi  4253  mptfvex  5199  rdgisucinc  5912  frecabex  5923
 Copyright terms: Public domain W3C validator