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Metamath Proof Explorer

Theorem opelopabf

Description: The law of concretion. Theorem 9.5 of Quine p. 61. This version of opelopab uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008)

		Ref	Expression
	Hypotheses	opelopabf.x	⊢ Ⅎ 𝑥 𝜓
		opelopabf.y	⊢ Ⅎ 𝑦 𝜒
		opelopabf.1	⊢ 𝐴 ∈ V
		opelopabf.2	⊢ 𝐵 ∈ V
		opelopabf.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
		opelopabf.4	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
	Assertion	opelopabf	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		opelopabf.x	⊢ Ⅎ 𝑥 𝜓
2		opelopabf.y	⊢ Ⅎ 𝑦 𝜒
3		opelopabf.1	⊢ 𝐴 ∈ V
4		opelopabf.2	⊢ 𝐵 ∈ V
5		opelopabf.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
6		opelopabf.4	⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
7		opelopabsb	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 )
8		nfcv	⊢ Ⅎ 𝑥 𝐵
9	8 1	nfsbcw	⊢ Ⅎ 𝑥 [ 𝐵 / 𝑦 ] 𝜓
10	5	sbcbidv	⊢ ( 𝑥 = 𝐴 → ( [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] 𝜓 ) )
11	9 10	sbciegf	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] 𝜓 ) )
12	3 11	ax-mp	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] 𝜓 )
13	2 6	sbciegf	⊢ ( 𝐵 ∈ V → ( [ 𝐵 / 𝑦 ] 𝜓 ↔ 𝜒 ) )
14	4 13	ax-mp	⊢ ( [ 𝐵 / 𝑦 ] 𝜓 ↔ 𝜒 )
15	7 12 14	3bitri	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜒 )