Theorem nfriota 7142

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 1811	. . 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 7137	. 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑))
7	6	mptru 1549	1 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑)

Syntax hints:  ⊤wtru 1543  Ⅎwnf 1790  Ⅎwnfc 2879  ℩crio 7128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-in 3850  df-ss 3860  df-sn 4517  df-uni 4797  df-iota 6297  df-riota 7129

This theorem is referenced by:  csbriota  7145  nfoi  9053  lble  11672  nosupbnd1  33562  noinfbnd1  33577  riotasvd  36615  riotasv2d  36616  riotasv2s  36617  cdleme26ee  38019  cdleme31sn1  38040  cdlemefs32sn1aw  38073  cdleme43fsv1snlem  38079  cdleme41sn3a  38092  cdleme32d  38103  cdleme32f  38105  cdleme40m  38126  cdleme40n  38127  cdlemk36  38572  cdlemk38  38574  cdlemkid  38595  cdlemk19x  38602  cdlemk11t  38605