Theorem sbim 1980

Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)

Assertion

Ref	Expression
sbim	|- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Proof of Theorem sbim

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbimv 1916	. . . 4 |- ( [ z / x ] ( ph -> ps ) <-> ( [ z / x ] ph -> [ z / x ] ps ) )
2	1	sbbii 1787	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / z ] ( [ z / x ] ph -> [ z / x ] ps ) )
3		sbimv 1916	. . 3 |- ( [ y / z ] ( [ z / x ] ph -> [ z / x ] ps ) <-> ( [ y / z ] [ z / x ] ph -> [ y / z ] [ z / x ] ps ) )
4	2, 3	bitri 184	. 2 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> ( [ y / z ] [ z / x ] ph -> [ y / z ] [ z / x ] ps ) )
5		ax-17 1548	. . 3 |- ( ( ph -> ps ) -> A. z ( ph -> ps ) )
6	5	sbco2vh 1972	. 2 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / x ] ( ph -> ps ) )
7		ax-17 1548	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2vh 1972	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1548	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2vh 1972	. . 3 |- ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps )
11	8, 10	imbi12i 239	. 2 |- ( ( [ y / z ] [ z / x ] ph -> [ y / z ] [ z / x ] ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
12	4, 6, 11	3bitr3i 210	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 105   [wsb 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785

This theorem is referenced by:  sbrim  1983  sblim  1984  sbbi  1986  moimv  2119  nfraldya  2540  sbcimg  3039  zfregfr  4621  tfi  4629  peano2  4642