Theorem nfsbd 1970

Description: Deduction version of nfsb 1939. (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 1512	. 2 |- ( ph -> A. x ph )
3		nfsbd.2	. . 3 |- ( ph -> F/ z ps )
4	3	alimi 1448	. 2 |- ( A. x ph -> A. x F/ z ps )
5		nfsbt 1969	. 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 1346   F/wnf 1453   [wsb 1755

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  nfeud  2035  nfabd  2332  nfraldya  2505  nfrexdya  2506  cbvrald  13823