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Theorem nfriota 5816
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 1459 . . 3 𝑦
2 nfriota.1 . . . 4 𝑥𝜑
32a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
4 nfriota.2 . . . 4 𝑥𝐴
54a1i 9 . . 3 (⊤ → 𝑥𝐴)
61, 3, 5nfriotadxy 5815 . 2 (⊤ → 𝑥(𝑦𝐴 𝜑))
76mptru 1357 1 𝑥(𝑦𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wtru 1349  wnf 1453  wnfc 2299  crio 5806
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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-sn 3587  df-uni 3795  df-iota 5158  df-riota 5807
This theorem is referenced by:  csbriotag  5819  lble  8856
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