Theorem sban 1889

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 1828	. . . 4 |- ( [ z / x ] ( ph /\ ps ) <-> ( [ z / x ] ph /\ [ z / x ] ps ) )
2	1	sbbii 1706	. . 3 |- ( [ y / z ] [ z / x ] ( ph /\ ps ) <-> [ y / z ] ( [ z / x ] ph /\ [ z / x ] ps ) )
3		sbanv 1828	. . 3 |- ( [ y / z ] ( [ z / x ] ph /\ [ z / x ] ps ) <-> ( [ y / z ] [ z / x ] ph /\ [ y / z ] [ z / x ] ps ) )
4	2, 3	bitri 183	. 2 |- ( [ y / z ] [ z / x ] ( ph /\ ps ) <-> ( [ y / z ] [ z / x ] ph /\ [ y / z ] [ z / x ] ps ) )
5		ax-17 1474	. . 3 |- ( ( ph /\ ps ) -> A. z ( ph /\ ps ) )
6	5	sbco2v 1881	. 2 |- ( [ y / z ] [ z / x ] ( ph /\ ps ) <-> [ y / x ] ( ph /\ ps ) )
7		ax-17 1474	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2v 1881	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1474	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2v 1881	. . 3 |- ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps )
11	8, 10	anbi12i 451	. 2 |- ( ( [ y / z ] [ z / x ] ph /\ [ y / z ] [ z / x ] ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )
12	4, 6, 11	3bitr3i 209	1 |- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   [wsb 1703

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704

This theorem is referenced by:  sb3an  1892  sbbi  1893  sbmo  2019  moanim  2034  sbabel  2266  nfrexdya  2429  cbvreu  2610  rmo3f  2834  sbcan  2903  sbcang  2904  rmo3  2952  inab  3291  difab  3292  exss  4087  inopab  4609  bdcriota  12662