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 → ( 𝜑 ↔ 𝜓 ) )
	eloprabg.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
	eloprabg.3	⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
Assertion	eloprabg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ↔ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		eloprabg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		eloprabg.2	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3		eloprabg.3	⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
4	1 2 3	syl3an9b	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜃 ) )
5	4	eloprabga	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ↔ 𝜃 ) )