Theorem sbcop

Description: The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of each of its components. (Contributed by AV, 8-Apr-2023)

	Ref	Expression
Hypothesis	sbcop.z	|- ( z = <. x , y >. -> ( ph <-> ps ) )
Assertion	sbcop	|- ( [. b / y ]. [. a / x ]. ps <-> [. <. a , b >. / z ]. ph )

Proof

Step	Hyp	Ref	Expression
1		sbcop.z	|- ( z = <. x , y >. -> ( ph <-> ps ) )
2	1	sbcop1	|- ( [. a / x ]. ps <-> [. <. a , y >. / z ]. ph )
3	2	sbcbii	|- ( [. b / y ]. [. a / x ]. ps <-> [. b / y ]. [. <. a , y >. / z ]. ph )
4		sbcnestgw	|- ( b e. _V -> ( [. b / y ]. [. <. a , y >. / z ]. ph <-> [. [_ b / y ]_ <. a , y >. / z ]. ph ) )
5	4	elv	|- ( [. b / y ]. [. <. a , y >. / z ]. ph <-> [. [_ b / y ]_ <. a , y >. / z ]. ph )
6		csbopg	|- ( b e. _V -> [_ b / y ]_ <. a , y >. = <. [_ b / y ]_ a , [_ b / y ]_ y >. )
7	6	elv	|- [_ b / y ]_ <. a , y >. = <. [_ b / y ]_ a , [_ b / y ]_ y >.
8		vex	|- b e. _V
9	8	csbconstgi	|- [_ b / y ]_ a = a
10	8	csbvargi	|- [_ b / y ]_ y = b
11	9 10	opeq12i	|- <. [_ b / y ]_ a , [_ b / y ]_ y >. = <. a , b >.
12	7 11	eqtri	|- [_ b / y ]_ <. a , y >. = <. a , b >.
13		dfsbcq	|- ( [_ b / y ]_ <. a , y >. = <. a , b >. -> ( [. [_ b / y ]_ <. a , y >. / z ]. ph <-> [. <. a , b >. / z ]. ph ) )
14	12 13	ax-mp	|- ( [. [_ b / y ]_ <. a , y >. / z ]. ph <-> [. <. a , b >. / z ]. ph )
15	3 5 14	3bitri	|- ( [. b / y ]. [. a / x ]. ps <-> [. <. a , b >. / z ]. ph )