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Metamath Proof Explorer

Theorem 3netr4d

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012) (Proof shortened by Wolf Lammen, 21-Nov-2019)

	Ref	Expression
Hypotheses	3netr4d.1	$⊢ φ \to A \neq B$
	3netr4d.2	$⊢ φ \to C = A$
	3netr4d.3	$⊢ φ \to D = B$
Assertion	3netr4d	$⊢ φ \to C \neq D$

Proof

Step	Hyp	Ref	Expression
1		3netr4d.1	$⊢ φ \to A \neq B$
2		3netr4d.2	$⊢ φ \to C = A$
3		3netr4d.3	$⊢ φ \to D = B$
4	2 1	eqnetrd	$⊢ φ \to C \neq B$
5	4 3	neeqtrrd	$⊢ φ \to C \neq D$