Theorem eqnetrrd 2386

Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)

Hypotheses

Ref	Expression
eqnetrrd.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqnetrrd.2	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Assertion

Ref	Expression
eqnetrrd	⊢ (𝜑 → 𝐵 ≠ 𝐶)

Proof of Theorem eqnetrrd

Step	Hyp	Ref	Expression
1		eqnetrrd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	eqcomd 2195	. 2 ⊢ (𝜑 → 𝐵 = 𝐴)
3		eqnetrrd.2	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
4	2, 3	eqnetrd 2384	1 ⊢ (𝜑 → 𝐵 ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1364   ≠ wne 2360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-ne 2361

This theorem is referenced by:  netap  7282  2omotaplemap  7285  pcadd  12371