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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 → 𝐴 ≠ 𝐵 )
	3netr3d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	3netr3d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	3netr3d	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3netr3d.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
2		3netr3d.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		3netr3d.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
4	1 3	neeqtrd	⊢ ( 𝜑 → 𝐴 ≠ 𝐷 )
5	2 4	eqnetrrd	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )