Theorem nfsbd 2111

Description: Deduction version of nfsb 2109. (Contributed by NM, 15-Feb-2013.)

Hypotheses

Ref	Expression
nfsbd.1	|- F/xph
nfsbd.2	|- (ph -> F/zps)

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/xph
2		nfsbd.2	. . . 4 |- (ph -> F/zps)
3	1, 2	alrimi 1765	. . 3 |- (ph -> A.x F/zps)
4		nfsb4t 2080	. . 3 |- (A.x F/zps -> (-. A.z z = y -> F/z[y / x]ps))
5	3, 4	syl 15	. 2 |- (ph -> (-. A.z z = y -> F/z[y / x]ps))
6		a16nf 2051	. 2 |- (A.z z = y -> F/z[y / x]ps)
7	5, 6	pm2.61d2 152	1 |- (ph -> F/z[y / x]ps)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540   F/wnf 1544  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  nfabd2  2508