Theorem sbim 1868

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 1814	. . . 4 |- ( [ z / x ] ( ph -> ps ) <-> ( [ z / x ] ph -> [ z / x ] ps ) )
2	1	sbbii 1688	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / z ] ( [ z / x ] ph -> [ z / x ] ps ) )
3		sbimv 1814	. . 3 |- ( [ y / z ] ( [ z / x ] ph -> [ z / x ] ps ) <-> ( [ y / z ] [ z / x ] ph -> [ y / z ] [ z / x ] ps ) )
4	2, 3	bitri 182	. 2 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> ( [ y / z ] [ z / x ] ph -> [ y / z ] [ z / x ] ps ) )
5		ax-17 1459	. . 3 |- ( ( ph -> ps ) -> A. z ( ph -> ps ) )
6	5	sbco2v 1862	. 2 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / x ] ( ph -> ps ) )
7		ax-17 1459	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2v 1862	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1459	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2v 1862	. . 3 |- ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps )
11	8, 10	imbi12i 237	. 2 |- ( ( [ y / z ] [ z / x ] ph -> [ y / z ] [ z / x ] ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
12	4, 6, 11	3bitr3i 208	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 103   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbrim  1871  sblim  1872  sbbi  1874  moimv  2007  nfraldya  2400  sbcimg  2855  zfregfr  4316  tfi  4323  peano2  4336