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Metamath Proof Explorer

Theorem mpodifsnif

Description: A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019)

		Ref	Expression
	Assertion	mpodifsnif	$⊢ (i \in (A ∖ \{X\}), j \in B ⟼ if (i = X, C, D)) = (i \in (A ∖ \{X\}), j \in B ⟼ D)$

Proof

Step	Hyp	Ref	Expression
1		eldifsnneq	$⊢ i \in (A ∖ \{X\}) \to \neg i = X$
2	1	adantr	$⊢ (i \in (A ∖ \{X\}) \land j \in B) \to \neg i = X$
3	2	iffalsed	$⊢ (i \in (A ∖ \{X\}) \land j \in B) \to if (i = X, C, D) = D$
4	3	mpoeq3ia	$⊢ (i \in (A ∖ \{X\}), j \in B ⟼ if (i = X, C, D)) = (i \in (A ∖ \{X\}), j \in B ⟼ D)$