Theorem sbim 1982

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 1918	. . . 4 |- ( [ z / x ] ( ph -> ps ) <-> ( [ z / x ] ph -> [ z / x ] ps ) )
2	1	sbbii 1789	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / z ] ( [ z / x ] ph -> [ z / x ] ps ) )
3		sbimv 1918	. . 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 1550	. . 3 |- ( ( ph -> ps ) -> A. z ( ph -> ps ) )
6	5	sbco2vh 1974	. 2 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / x ] ( ph -> ps ) )
7		ax-17 1550	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2vh 1974	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1550	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2vh 1974	. . 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 1786

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  sbrim  1985  sblim  1986  sbbi  1988  moimv  2122  nfraldya  2543  sbcimg  3047  zfregfr  4640  tfi  4648  peano2  4661