Theorem nfsbd 2442

Description: Deduction version of nfsb 2440. (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	. . . 4 |- F/ x ph
2		nfsbd.2	. . . 4 |- ( ph -> F/ z ps )
3	1, 2	alrimi 2082	. . 3 |- ( ph -> A. x F/ z ps )
4		nfsb4t 2389	. . 3 |- ( A. x F/ z ps -> ( -. A. z z = y -> F/ z [ y / x ] ps ) )
5	3, 4	syl 17	. 2 |- ( ph -> ( -. A. z z = y -> F/ z [ y / x ] ps ) )
6		axc16nf 2137	. 2 |- ( A. z z = y -> F/ z [ y / x ] ps )
7	5, 6	pm2.61d2 172	1 |- ( ph -> F/ z [ y / x ] ps )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  nfabd2  2784  wl-sb8eut  33359