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Theorem nfriota 5477
 Description: A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.)
Hypotheses
Ref Expression
nfriota.1 𝑥𝜑
nfriota.2 𝑥𝐴
Assertion
Ref Expression
nfriota 𝑥(𝑦𝐴 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfriota
StepHypRef Expression
1 nftru 1355 . . 3 𝑦
2 nfriota.1 . . . 4 𝑥𝜑
32a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
4 nfriota.2 . . . 4 𝑥𝐴
54a1i 9 . . 3 (⊤ → 𝑥𝐴)
61, 3, 5nfriotadxy 5476 . 2 (⊤ → 𝑥(𝑦𝐴 𝜑))
76trud 1252 1 𝑥(𝑦𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:  ⊤wtru 1244  Ⅎwnf 1349  Ⅎwnfc 2165  ℩crio 5467 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 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-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-sn 3381  df-uni 3581  df-iota 4867  df-riota 5468 This theorem is referenced by:  csbriotag  5480
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