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Theorem nfriota 7115
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 1796 . . 3 𝑦
2 nfriota.1 . . . 4 𝑥𝜑
32a1i 11 . . 3 (⊤ → Ⅎ𝑥𝜑)
4 nfriota.2 . . . 4 𝑥𝐴
54a1i 11 . . 3 (⊤ → 𝑥𝐴)
61, 3, 5nfriotadw 7111 . 2 (⊤ → 𝑥(𝑦𝐴 𝜑))
76mptru 1535 1 𝑥(𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wtru 1529  wnf 1775  wnfc 2958  crio 7102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-sn 4558  df-uni 4831  df-iota 6307  df-riota 7103
This theorem is referenced by:  csbriota  7118  nfoi  8966  lble  11581  nosupbnd1  33111  riotasvd  35972  riotasv2d  35973  riotasv2s  35974  cdleme26ee  37376  cdleme31sn1  37397  cdlemefs32sn1aw  37430  cdleme43fsv1snlem  37436  cdleme41sn3a  37449  cdleme32d  37460  cdleme32f  37462  cdleme40m  37483  cdleme40n  37484  cdlemk36  37929  cdlemk38  37931  cdlemkid  37952  cdlemk19x  37959  cdlemk11t  37962
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