Theorem sbc3an

Description: Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006) (Revised by NM, 17-Aug-2018)

	Ref	Expression
Assertion	sbc3an	|- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) )

Proof

Step	Hyp	Ref	Expression
1		sbcan	|- ( [. A / x ]. ( ( ph /\ ps ) /\ ch ) <-> ( [. A / x ]. ( ph /\ ps ) /\ [. A / x ]. ch ) )
2		sbcan	|- ( [. A / x ]. ( ph /\ ps ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps ) )
3	1 2	bianbi	|- ( [. A / x ]. ( ( ph /\ ps ) /\ ch ) <-> ( ( [. A / x ]. ph /\ [. A / x ]. ps ) /\ [. A / x ]. ch ) )
4		df-3an	|- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
5	4	sbcbii	|- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> [. A / x ]. ( ( ph /\ ps ) /\ ch ) )
6		df-3an	|- ( ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph /\ [. A / x ]. ps ) /\ [. A / x ]. ch ) )
7	3 5 6	3bitr4i	|- ( [. A / x ]. ( ph /\ ps /\ ch ) <-> ( [. A / x ]. ph /\ [. A / x ]. ps /\ [. A / x ]. ch ) )