Theorem sban 1983

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 1913	. . . 4 |- ( [ z / x ] ( ph /\ ps ) <-> ( [ z / x ] ph /\ [ z / x ] ps ) )
2	1	sbbii 1788	. . 3 |- ( [ y / z ] [ z / x ] ( ph /\ ps ) <-> [ y / z ] ( [ z / x ] ph /\ [ z / x ] ps ) )
3		sbanv 1913	. . 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 1549	. . 3 |- ( ( ph /\ ps ) -> A. z ( ph /\ ps ) )
6	5	sbco2vh 1973	. 2 |- ( [ y / z ] [ z / x ] ( ph /\ ps ) <-> [ y / x ] ( ph /\ ps ) )
7		ax-17 1549	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2vh 1973	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1549	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2vh 1973	. . 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 1785

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by:  sb3an  1986  sbbi  1987  sbmo  2113  moanim  2128  sbabel  2375  nfrexdya  2542  cbvreu  2736  rmo3f  2970  sbcan  3041  sbcang  3042  rmo3  3090  inab  3441  difab  3442  exss  4271  inopab  4810  bdcriota  15819