Theorem nfriota 7105

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 1806	. . 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 7101	. 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑))
7	6	mptru 1545	1 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑)

Syntax hints:  ⊤wtru 1539  Ⅎwnf 1785  Ⅎwnfc 2936  ℩crio 7092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-sn 4526  df-uni 4801  df-iota 6283  df-riota 7093

This theorem is referenced by:  csbriota  7108  nfoi  8962  lble  11580  nosupbnd1  33327  riotasvd  36252  riotasv2d  36253  riotasv2s  36254  cdleme26ee  37656  cdleme31sn1  37677  cdlemefs32sn1aw  37710  cdleme43fsv1snlem  37716  cdleme41sn3a  37729  cdleme32d  37740  cdleme32f  37742  cdleme40m  37763  cdleme40n  37764  cdlemk36  38209  cdlemk38  38211  cdlemkid  38232  cdlemk19x  38239  cdlemk11t  38242