Theorem eloprabg

Description: The law of concretion for operation class abstraction. Compare elopab . (Contributed by NM, 14-Sep-1999) (Revised by David Abernethy, 19-Jun-2012)

	Ref	Expression
Hypotheses	eloprabg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
	eloprabg.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
	eloprabg.3	$$⊢z=C\to \left(\chi \leftrightarrow \theta \right)$$
Assertion	eloprabg	$$⊢\left(A\in V\wedge B\in W\wedge C\in X\right)\to \left(⟨⟨A,B⟩,C⟩\in \left\{⟨⟨x,y⟩,z⟩|\phi \right\}\leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		eloprabg.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2		eloprabg.2	$$⊢y=B\to \left(\psi \leftrightarrow \chi \right)$$
3		eloprabg.3	$$⊢z=C\to \left(\chi \leftrightarrow \theta \right)$$
4	1 2 3	syl3an9b	$$⊢\left(x=A\wedge y=B\wedge z=C\right)\to \left(\phi \leftrightarrow \theta \right)$$
5	4	eloprabga	$$⊢\left(A\in V\wedge B\in W\wedge C\in X\right)\to \left(⟨⟨A,B⟩,C⟩\in \left\{⟨⟨x,y⟩,z⟩|\phi \right\}\leftrightarrow \theta \right)$$