Theorem sban 2008

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

Assertion

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

Proof of Theorem sban

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbanv 1938	. . . 4 |- ( [ z / x ] ( ph /\ ps ) <-> ( [ z / x ] ph /\ [ z / x ] ps ) )
2	1	sbbii 1813	. . 3 |- ( [ y / z ] [ z / x ] ( ph /\ ps ) <-> [ y / z ] ( [ z / x ] ph /\ [ z / x ] ps ) )
3		sbanv 1938	. . 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 1574	. . 3 |- ( ( ph /\ ps ) -> A. z ( ph /\ ps ) )
6	5	sbco2vh 1998	. 2 |- ( [ y / z ] [ z / x ] ( ph /\ ps ) <-> [ y / x ] ( ph /\ ps ) )
7		ax-17 1574	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2vh 1998	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1574	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2vh 1998	. . 3 |- ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps )
11	8, 10	anbi12i 460	. 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:   /\ wa 104   <-> wb 105   [wsb 1810

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  sb3an  2011  sbbi  2012  sbmo  2139  moanim  2154  sbabel  2401  nfrexdya  2568  cbvreu  2765  rmo3f  3003  sbcan  3074  sbcang  3075  rmo3  3124  inab  3475  difab  3476  exss  4319  inopab  4862  bdcriota  16478