Theorem 3netr4g

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012)

	Ref	Expression
Hypotheses	3netr4g.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
	3netr4g.2	⊢ 𝐶 = 𝐴
	3netr4g.3	⊢ 𝐷 = 𝐵
Assertion	3netr4g	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3netr4g.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
2		3netr4g.2	⊢ 𝐶 = 𝐴
3		3netr4g.3	⊢ 𝐷 = 𝐵
4	2 3	neeq12i	⊢ ( 𝐶 ≠ 𝐷 ↔ 𝐴 ≠ 𝐵 )
5	1 4	sylibr	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )