Theorem nfsbd 1948

Description: Deduction version of nfsb 1917. (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 1499	. 2 |- ( ph -> A. x ph )
3		nfsbd.2	. . 3 |- ( ph -> F/ z ps )
4	3	alimi 1431	. 2 |- ( A. x ph -> A. x F/ z ps )
5		nfsbt 1947	. 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 1329   F/wnf 1436   [wsb 1735

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

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  nfeud  2013  nfabd  2298  nfraldya  2467  nfrexdya  2468  cbvrald  12984