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

Theorem nfif 3577
Description: Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
nfif.1 𝑥𝜑
nfif.2 𝑥𝐴
nfif.3 𝑥𝐵
Assertion
Ref Expression
nfif 𝑥if(𝜑, 𝐴, 𝐵)

Proof of Theorem nfif
StepHypRef Expression
1 nfif.1 . . . 4 𝑥𝜑
21a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
3 nfif.2 . . . 4 𝑥𝐴
43a1i 9 . . 3 (⊤ → 𝑥𝐴)
5 nfif.3 . . . 4 𝑥𝐵
65a1i 9 . . 3 (⊤ → 𝑥𝐵)
72, 4, 6nfifd 3576 . 2 (⊤ → 𝑥if(𝜑, 𝐴, 𝐵))
87mptru 1373 1 𝑥if(𝜑, 𝐴, 𝐵)
Colors of variables: wff set class
Syntax hints:  wtru 1365  wnf 1471  wnfc 2319  ifcif 3549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-if 3550
This theorem is referenced by:  nfsum1  11396  nfsum  11397  sumrbdclem  11417  summodclem2a  11421  zsumdc  11424  fsum3  11427  isumss  11431  isumss2  11433  fsum3cvg2  11434  nfcprod1  11594  nfcprod  11595  cbvprod  11598  prodrbdclem  11611  prodmodclem2a  11616  zproddc  11619  fprodseq  11623  fprodntrivap  11624  prodssdc  11629  pcmpt  12375  pcmptdvds  12377
  Copyright terms: Public domain W3C validator