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

Theorem spcgf 2629
 Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)
Hypotheses
Ref Expression
spcgf.1 xA
spcgf.2 xψ
spcgf.3 (x = A → (φψ))
Assertion
Ref Expression
spcgf (A 𝑉 → (xφψ))

Proof of Theorem spcgf
StepHypRef Expression
1 spcgf.2 . . 3 xψ
2 spcgf.1 . . 3 xA
31, 2spcgft 2624 . 2 (x(x = A → (φψ)) → (A 𝑉 → (xφψ)))
4 spcgf.3 . 2 (x = A → (φψ))
53, 4mpg 1337 1 (A 𝑉 → (xφψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  Ⅎwnfc 2162 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-bndl 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:  spcgv  2634  rspc  2644  elabgt  2678  eusvnf  4151  mpt2fvex  5771
 Copyright terms: Public domain W3C validator