Theorem nfsbd 1964

Description: Deduction version of nfsb 1933. (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 1506	. 2 |- ( ph -> A. x ph )
3		nfsbd.2	. . 3 |- ( ph -> F/ z ps )
4	3	alimi 1442	. 2 |- ( A. x ph -> A. x F/ z ps )
5		nfsbt 1963	. 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 1340   F/wnf 1447   [wsb 1749

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750

This theorem is referenced by:  nfeud  2029  nfabd  2326  nfraldya  2499  nfrexdya  2500  cbvrald  13504