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Metamath Proof Explorer

Theorem 3netr3d

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012) (Proof shortened by Wolf Lammen, 19-Nov-2019)

	Ref	Expression
Hypotheses	3netr3d.1	$⊢ φ \to A \neq B$
	3netr3d.2	$⊢ φ \to A = C$
	3netr3d.3	$⊢ φ \to B = D$
Assertion	3netr3d	$⊢ φ \to C \neq D$

Proof

Step	Hyp	Ref	Expression
1		3netr3d.1	$⊢ φ \to A \neq B$
2		3netr3d.2	$⊢ φ \to A = C$
3		3netr3d.3	$⊢ φ \to B = D$
4	1 3	neeqtrd	$⊢ φ \to A \neq D$
5	2 4	eqnetrrd	$⊢ φ \to C \neq D$