Theorem nfriota 7338

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 1804	. . 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 7334	. 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑))
7	6	mptru 1547	1 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜑)

Syntax hints:  ⊤wtru 1541  Ⅎwnf 1783  Ⅎwnfc 2876  ℩crio 7325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-v 3446  df-ss 3928  df-sn 4586  df-uni 4868  df-iota 6452  df-riota 7326

This theorem is referenced by:  csbriota  7341  nfoi  9443  lble  12111  nosupbnd1  27659  noinfbnd1  27674  riotasvd  38942  riotasv2d  38943  riotasv2s  38944  cdleme26ee  40347  cdleme31sn1  40368  cdlemefs32sn1aw  40401  cdleme43fsv1snlem  40407  cdleme41sn3a  40420  cdleme32d  40431  cdleme32f  40433  cdleme40m  40454  cdleme40n  40455  cdlemk36  40900  cdlemk38  40902  cdlemkid  40923  cdlemk19x  40930  cdlemk11t  40933