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 1575	. . 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 1575	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2vh 1998	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1575	. . . 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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811

This theorem is referenced by:  sb3an  2011  sbbi  2012  sbmo  2139  moanim  2154  sbabel  2402  nfrexdya  2569  cbvreu  2766  rmo3f  3004  sbcan  3075  sbcang  3076  rmo3  3125  inab  3477  difab  3478  exss  4325  inopab  4868  bdcriota  16582