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

Theorem nfsbc 2861
 Description: Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
nfsbc.1 𝑥𝐴
nfsbc.2 𝑥𝜑
Assertion
Ref Expression
nfsbc 𝑥[𝐴 / 𝑦]𝜑

Proof of Theorem nfsbc
StepHypRef Expression
1 nftru 1401 . . 3 𝑦
2 nfsbc.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfsbc.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfsbcd 2860 . 2 (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
76mptru 1299 1 𝑥[𝐴 / 𝑦]𝜑
 Colors of variables: wff set class Syntax hints:  ⊤wtru 1291  Ⅎwnf 1395  Ⅎwnfc 2216  [wsbc 2841 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-sbc 2842 This theorem is referenced by:  cbvralcsf  2991  cbvrexcsf  2992  opelopabf  4110  ralrnmpt  5455  rexrnmpt  5456  dfopab2  5973  dfoprab3s  5974  mpt2xopoveq  6019
 Copyright terms: Public domain W3C validator