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Metamath Proof Explorer

Theorem iftrue

Description: Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999) (Proof shortened by Andrew Salmon, 26-Jun-2011)

		Ref	Expression
	Assertion	iftrue	$⊢ φ \to if (φ, A, B) = A$

Proof

Step	Hyp	Ref	Expression
1		dfif2	$⊢ if (φ, A, B) = \{x \| ((x \in B \to φ) \to (x \in A \land φ))\}$
2		dedlem0a	$⊢ φ \to (x \in A \leftrightarrow ((x \in B \to φ) \to (x \in A \land φ)))$
3	2	eqabdv	$⊢ φ \to A = \{x \| ((x \in B \to φ) \to (x \in A \land φ))\}$
4	1 3	eqtr4id	$⊢ φ \to if (φ, A, B) = A$