This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem disjeq1d

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016)

		Ref	Expression
	Hypothesis	disjeq1d.1	$⊢ φ \to A = B$
	Assertion	disjeq1d	$⊢ φ \to (\underset{x \in A}{Disj} C \leftrightarrow \underset{x \in B}{Disj} C)$

Step	Hyp	Ref	Expression
1		disjeq1d.1	$⊢ φ \to A = B$
2		disjeq1	$⊢ A = B \to (\underset{x \in A}{Disj} C \leftrightarrow \underset{x \in B}{Disj} C)$
3	1 2	syl	$⊢ φ \to (\underset{x \in A}{Disj} C \leftrightarrow \underset{x \in B}{Disj} C)$