Theorem nfriota 5964

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 1512	. . 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 5963	. 2 |- ( T. -> F/_ x ( iota_ y e. A ph ) )
7	6	mptru 1404	1 |- F/_ x ( iota_ y e. A ph )

Syntax hints:   T. wtru 1396   F/wnf 1506   F/_wnfc 2359   iota_crio 5953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-sn 3672  df-uni 3889  df-iota 5278  df-riota 5954

This theorem is referenced by:  csbriotag  5968  lble  9094