Theorem sban 2015

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 1944	. . . 4 |- ( [ z / x ] ( ph /\ ps ) <-> ( [ z / x ] ph /\ [ z / x ] ps ) )
2	1	sbbii 1818	. . 3 |- ( [ y / z ] [ z / x ] ( ph /\ ps ) <-> [ y / z ] ( [ z / x ] ph /\ [ z / x ] ps ) )
3		sbanv 1944	. . 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 1579	. . 3 |- ( ( ph /\ ps ) -> A. z ( ph /\ ps ) )
6	5	sbco2vh 2005	. 2 |- ( [ y / z ] [ z / x ] ( ph /\ ps ) <-> [ y / x ] ( ph /\ ps ) )
7		ax-17 1579	. . . 4 |- ( ph -> A. z ph )
8	7	sbco2vh 2005	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9		ax-17 1579	. . . 4 |- ( ps -> A. z ps )
10	9	sbco2vh 2005	. . 3 |- ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps )
11	8, 10	anbi12i 464	. 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 1815

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sb3an  2018  sbbi  2019  sbmo  2146  moanim  2161  sbabel  2419  nfrexdya  2586  cbvreu  2784  rmo3f  3023  sbcan  3094  sbcang  3095  rmo3  3144  inab  3499  difab  3500  exss  4362  inopab  4907  bdcriota  16823