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Theorem nfif 3356
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 3355 . 2 (⊤ → 𝑥if(𝜑, 𝐴, 𝐵))
87trud 1252 1 𝑥if(𝜑, 𝐴, 𝐵)
Colors of variables: wff set class
Syntax hints:  wtru 1244  wnf 1349  wnfc 2165  ifcif 3331
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-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-if 3332
This theorem is referenced by:  nfsum1  9848  nfsum  9849
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