Theorem sbim 1965

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 1905	. . . 4 |- ( [ z / x ] ( ph -> ps ) <-> ( [ z / x ] ph -> [ z / x ] ps ) )
2	1	sbbii 1776	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / z ] ( [ z / x ] ph -> [ z / x ] ps ) )
3		sbimv 1905	. . 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 1537	. . 3 |- ( ( ph -> ps ) -> A. z ( ph -> ps ) )
6	5	sbco2vh 1957	. 2 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / x ] ( ph -> ps ) )
7		ax-17 1537	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2vh 1957	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1537	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2vh 1957	. . 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 1773

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sbrim  1968  sblim  1969  sbbi  1971  moimv  2104  nfraldya  2525  sbcimg  3019  zfregfr  4591  tfi  4599  peano2  4612