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Metamath Proof Explorer

Theorem opelopabg

Description: The law of concretion. Theorem 9.5 of Quine p. 61. (Contributed by NM, 28-May-1995) (Revised by Mario Carneiro, 19-Dec-2013)

	Ref	Expression
Hypotheses	opelopabg.1	$⊢ x = A \to (φ \leftrightarrow ψ)$
	opelopabg.2	$⊢ y = B \to (ψ \leftrightarrow χ)$
Assertion	opelopabg	$⊢ (A \in V \land B \in W) \to (⟨A, B⟩ \in \{⟨x, y⟩ \| φ\} \leftrightarrow χ)$

Step	Hyp	Ref	Expression
1		opelopabg.1	$⊢ x = A \to (φ \leftrightarrow ψ)$
2		opelopabg.2	$⊢ y = B \to (ψ \leftrightarrow χ)$
3	1 2	sylan9bb	$⊢ (x = A \land y = B) \to (φ \leftrightarrow χ)$
4	3	opelopabga	$⊢ (A \in V \land B \in W) \to (⟨A, B⟩ \in \{⟨x, y⟩ \| φ\} \leftrightarrow χ)$