Theorem nfriotadxy 5852

Description: Deduction version of nfriota 5853. (Contributed by Jim Kingdon, 12-Jan-2019.)

Hypotheses

Ref	Expression
nfriotadxy.1	|- F/ y ph
nfriotadxy.2	|- ( ph -> F/ x ps )
nfriotadxy.3	|- ( ph -> F/_ x A )

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem nfriotadxy

Step	Hyp	Ref	Expression
1		df-riota 5844	. 2 |- ( iota_ y e. A ps ) = ( iota y ( y e. A /\ ps ) )
2		nfriotadxy.1	. . 3 |- F/ y ph
3		nfcv 2329	. . . . . 6 |- F/_ x y
4	3	a1i 9	. . . . 5 |- ( ph -> F/_ x y )
5		nfriotadxy.3	. . . . 5 |- ( ph -> F/_ x A )
6	4, 5	nfeld 2345	. . . 4 |- ( ph -> F/ x y e. A )
7		nfriotadxy.2	. . . 4 |- ( ph -> F/ x ps )
8	6, 7	nfand 1578	. . 3 |- ( ph -> F/ x ( y e. A /\ ps ) )
9	2, 8	nfiotadw 5193	. 2 |- ( ph -> F/_ x ( iota y ( y e. A /\ ps ) ) )
10	1, 9	nfcxfrd 2327	1 |- ( ph -> F/_ x ( iota_ y e. A ps ) )

Syntax hints:   -> wi 4   /\ wa 104   F/wnf 1470   e. wcel 2158   F/_wnfc 2316   iotacio 5188   iota_crio 5843

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-sn 3610  df-uni 3822  df-iota 5190  df-riota 5844

This theorem is referenced by:  nfriota  5853