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Theorem nfif 3470
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 3469 . 2 (⊤ → 𝑥if(𝜑, 𝐴, 𝐵))
87mptru 1325 1 𝑥if(𝜑, 𝐴, 𝐵)
Colors of variables: wff set class
Syntax hints:  wtru 1317  wnf 1421  wnfc 2245  ifcif 3444
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-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-if 3445
This theorem is referenced by:  nfsum1  11093  nfsum  11094  sumrbdclem  11113  summodclem2a  11118  zsumdc  11121  fsum3  11124  isumss  11128  isumss2  11130  fsum3cvg2  11131
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