Theorem 3netr4g 2375

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

Hypotheses

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

Assertion

Ref	Expression
3netr4g	⊢ (𝜑 → 𝐶 ≠ 𝐷)

Proof of Theorem 3netr4g

Step	Hyp	Ref	Expression
1		3netr4g.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		3netr4g.2	. . 3 ⊢ 𝐶 = 𝐴
3		3netr4g.3	. . 3 ⊢ 𝐷 = 𝐵
4	2, 3	neeq12i 2357	. 2 ⊢ (𝐶 ≠ 𝐷 ↔ 𝐴 ≠ 𝐵)
5	1, 4	sylibr 133	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:   → wi 4   = wceq 1348   ≠ wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-ne 2341

This theorem is referenced by: (None)