Theorem hbsbd 1957

Description: Deduction version of hbsb 1922. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)

Hypotheses

Ref	Expression
hbsbd.1	|- ( ph -> A. x ph )
hbsbd.2	|- ( ph -> A. z ph )
hbsbd.3	|- ( ph -> ( ps -> A. z ps ) )

Assertion

Ref	Expression
hbsbd	|- ( ph -> ( [ y / x ] ps -> A. z [ y / x ] ps ) )

Distinct variable group:   y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)

Proof of Theorem hbsbd

Step	Hyp	Ref	Expression
1		hbsbd.2	. . . 4 |- ( ph -> A. z ph )
2	1	nfi 1438	. . 3 |- F/ z ph
3		hbsbd.3	. . . . . . 7 |- ( ph -> ( ps -> A. z ps ) )
4	1, 3	nfdh 1504	. . . . . 6 |- ( ph -> F/ z ps )
5	2, 4	nfim1 1550	. . . . 5 |- F/ z ( ph -> ps )
6	5	nfsb 1919	. . . 4 |- F/ z [ y / x ] ( ph -> ps )
7		hbsbd.1	. . . . . 6 |- ( ph -> A. x ph )
8	7	sbrim 1929	. . . . 5 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )
9	8	nfbii 1449	. . . 4 |- ( F/ z [ y / x ] ( ph -> ps ) <-> F/ z ( ph -> [ y / x ] ps ) )
10	6, 9	mpbi 144	. . 3 |- F/ z ( ph -> [ y / x ] ps )
11	2, 10	nfrimi 1505	. 2 |- ( ph -> F/ z [ y / x ] ps )
12	11	nfrd 1500	1 |- ( ph -> ( [ y / x ] ps -> A. 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: (None)