Theorem sbanOLD

Description: Obsolete version of sban as of 13-Aug-2023. Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sbn	|- ( [ y / x ] -. ( ph -> -. ps ) <-> -. [ y / x ] ( ph -> -. ps ) )
2		sbim	|- ( [ y / x ] ( ph -> -. ps ) <-> ( [ y / x ] ph -> [ y / x ] -. ps ) )
3		sbn	|- ( [ y / x ] -. ps <-> -. [ y / x ] ps )
4	3	imbi2i	|- ( ( [ y / x ] ph -> [ y / x ] -. ps ) <-> ( [ y / x ] ph -> -. [ y / x ] ps ) )
5	2 4	bitri	|- ( [ y / x ] ( ph -> -. ps ) <-> ( [ y / x ] ph -> -. [ y / x ] ps ) )
6	1 5	xchbinx	|- ( [ y / x ] -. ( ph -> -. ps ) <-> -. ( [ y / x ] ph -> -. [ y / x ] ps ) )
7		df-an	|- ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) )
8	7	sbbii	|- ( [ y / x ] ( ph /\ ps ) <-> [ y / x ] -. ( ph -> -. ps ) )
9		df-an	|- ( ( [ y / x ] ph /\ [ y / x ] ps ) <-> -. ( [ y / x ] ph -> -. [ y / x ] ps ) )
10	6 8 9	3bitr4i	|- ( [ y / x ] ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) )