Theorem nfsbd 1851

Description: Deduction version of nfsb 1822. (Contributed by NM, 15-Feb-2013.)

Hypotheses

Ref	Expression
nfsbd.1	|- F/ x ph
nfsbd.2	|- ( ph -> F/ z ps )

Assertion

Ref	Expression
nfsbd	|- ( ph -> F/ z [ y / x ] ps )

Distinct variable group:   y, z

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

Proof of Theorem nfsbd

Step	Hyp	Ref	Expression
1		nfsbd.1	. . 3 |- F/ x ph
2	1	nfri 1412	. 2 |- ( ph -> A. x ph )
3		nfsbd.2	. . 3 |- ( ph -> F/ z ps )
4	3	alimi 1344	. 2 |- ( A. x ph -> A. x F/ z ps )
5		nfsbt 1850	. 2 |- ( A. x F/ z ps -> F/ z [ y / x ] ps )
6	2, 4, 5	3syl 17	1 |- ( ph -> F/ z [ y / x ] ps )

Syntax hints:   -> wi 4   A.wal 1241   F/wnf 1349   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  nfeud  1916  nfabd  2196  nfraldya  2358  nfrexdya  2359  cbvrald  9927