Theorem nfriota 5739

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	|- F/ x ph
nfriota.2	|- F/_ x A

Assertion

Ref	Expression
nfriota	|- F/_ x ( iota_ y e. A ph )

Distinct variable group:   x, y

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

Proof of Theorem nfriota

Step	Hyp	Ref	Expression
1		nftru 1442	. . 3 |- F/ y T.
2		nfriota.1	. . . 4 |- F/ x ph
3	2	a1i 9	. . 3 |- ( T. -> F/ x ph )
4		nfriota.2	. . . 4 |- F/_ x A
5	4	a1i 9	. . 3 |- ( T. -> F/_ x A )
6	1, 3, 5	nfriotadxy 5738	. 2 |- ( T. -> F/_ x ( iota_ y e. A ph ) )
7	6	mptru 1340	1 |- F/_ x ( iota_ y e. A ph )

Syntax hints:   T. wtru 1332   F/wnf 1436   F/_wnfc 2268   iota_crio 5729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-sn 3533  df-uni 3737  df-iota 5088  df-riota 5730

This theorem is referenced by:  csbriotag  5742  lble  8705