Theorem sbim 1827

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 1773	. . . 4 |- ( [ z / x ] ( ph -> ps ) <-> ( [ z / x ] ph -> [ z / x ] ps ) )
2	1	sbbii 1648	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / z ] ( [ z / x ] ph -> [ z / x ] ps ) )
3		sbimv 1773	. . 3 |- ( [ y / z ] ( [ z / x ] ph -> [ z / x ] ps ) <-> ( [ y / z ] [ z / x ] ph -> [ y / z ] [ z / x ] ps ) )
4	2, 3	bitri 173	. 2 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> ( [ y / z ] [ z / x ] ph -> [ y / z ] [ z / x ] ps ) )
5		ax-17 1419	. . 3 |- ( ( ph -> ps ) -> A. z ( ph -> ps ) )
6	5	sbco2v 1821	. 2 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / x ] ( ph -> ps ) )
7		ax-17 1419	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2v 1821	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1419	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2v 1821	. . 3 |- ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps )
11	8, 10	imbi12i 228	. 2 |- ( ( [ y / z ] [ z / x ] ph -> [ y / z ] [ z / x ] ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
12	4, 6, 11	3bitr3i 199	1 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 98   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbrim  1830  sblim  1831  sbbi  1833  moimv  1966  nfraldya  2358  sbcimg  2804  zfregfr  4298  tfi  4305  peano2  4318