Theorem nfiotadxy 5049

Description: Deduction version of nfiotaxy 5050. (Contributed by Jim Kingdon, 21-Dec-2018.)

Hypotheses

Ref	Expression
nfiotadxy.1	|- F/ y ph
nfiotadxy.2	|- ( ph -> F/ x ps )

Assertion

Ref	Expression
nfiotadxy	|- ( ph -> F/_ x ( iota y ps ) )

Distinct variable group:   x, y

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

Proof of Theorem nfiotadxy

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 5047	. 2 |- ( iota y ps ) = U. { z | A. y ( ps <-> y = z ) }
2		nfv 1491	. . . 4 |- F/ z ph
3		nfiotadxy.1	. . . . 5 |- F/ y ph
4		nfiotadxy.2	. . . . . 6 |- ( ph -> F/ x ps )
5		nfcv 2255	. . . . . . . 8 |- F/_ x y
6		nfcv 2255	. . . . . . . 8 |- F/_ x z
7	5, 6	nfeq 2263	. . . . . . 7 |- F/ x y = z
8	7	a1i 9	. . . . . 6 |- ( ph -> F/ x y = z )
9	4, 8	nfbid 1550	. . . . 5 |- ( ph -> F/ x ( ps <-> y = z ) )
10	3, 9	nfald 1716	. . . 4 |- ( ph -> F/ x A. y ( ps <-> y = z ) )
11	2, 10	nfabd 2274	. . 3 |- ( ph -> F/_ x { z | A. y ( ps <-> y = z ) } )
12	11	nfunid 3709	. 2 |- ( ph -> F/_ x U. { z | A. y ( ps <-> y = z ) } )
13	1, 12	nfcxfrd 2253	1 |- ( ph -> F/_ x ( iota y ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1312   = wceq 1314   F/wnf 1419   {cab 2101   F/_wnfc 2242   U.cuni 3702   iotacio 5044

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-sn 3499  df-uni 3703  df-iota 5046

This theorem is referenced by:  nfiotaxy  5050  nfriotadxy  5692