Theorem nfriota 7381

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	⊢ Ⅎ𝑥𝜑
nfriota.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfriota	⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥, 𝑦)

Proof of Theorem nfriota

Step	Hyp	Ref	Expression
1		nftru 1833	. . 3 ⊢ Ⅎ𝑦⊤
2		nfriota.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4		nfriota.2	. . . 4 ⊢ Ⅎ𝑥𝐴
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
6	1, 3, 5	nfriotadw 7377	. 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑))
7	6	mptru 1576	1 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:  ⊤wtru 1570  Ⅎwnf 1812  Ⅎwnfc 2909  ℩crio 7368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-v 3456  df-ss 3921  df-sn 4589  df-uni 4872  df-iota 6492  df-riota 7369

This theorem is used by:  csbriota  7384  nfoi  9474  lble  12173  nosupbnd1  27889  noinfbnd1  27904  riotasvd  39758  riotasv2d  39759  riotasv2s  39760  cdleme26ee  41162  cdleme31sn1  41183  cdlemefs32sn1aw  41216  cdleme43fsv1snlem  41222  cdleme41sn3a  41235  cdleme32d  41246  cdleme32f  41248  cdleme40m  41269  cdleme40n  41270  cdlemk36  41715  cdlemk38  41717  cdlemkid  41738  cdlemk19x  41745  cdlemk11t  41748