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Metamath Proof Explorer

Theorem imadisjlnd

Description: Deduction form of one negated side of imadisj . (Contributed by Stanislas Polu, 9-Mar-2020)

		Ref	Expression
	Hypothesis	imadisjlnd.1	$⊢ φ \to dom A \cap B \neq \emptyset$
	Assertion	imadisjlnd	$⊢ φ \to A [B] \neq \emptyset$

Step	Hyp	Ref	Expression
1		imadisjlnd.1	$⊢ φ \to dom A \cap B \neq \emptyset$
2		imadisj	$⊢ A [B] = \emptyset \leftrightarrow dom A \cap B = \emptyset$
3	2	biimpi	$⊢ A [B] = \emptyset \to dom A \cap B = \emptyset$
4	3	necon3i	$⊢ dom A \cap B \neq \emptyset \to A [B] \neq \emptyset$
5	1 4	syl	$⊢ φ \to A [B] \neq \emptyset$