Theorem neeqtrd 2851

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

Hypotheses

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

Assertion

Ref	Expression
neeqtrd	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem neeqtrd

Step	Hyp	Ref	Expression
1		neeqtrd.1	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
2		neeqtrd.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐶)
3	2	neeq2d 2842	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶))
4	1, 3	mpbid 221	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   = wceq 1475   ≠ wne 2780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-cleq 2603  df-ne 2782

This theorem is referenced by:  neeqtrrd  2856  3netr3d  2858  xaddass2  11952  xov1plusxeqvd  12189  issubdrg  18628  ply1scln0  19482  qsssubdrg  19624  alexsublem  21658  cphsubrglem  22785  cphreccllem  22786  mdegldg  23630  tglinethru  25331  footex  25413  eupath2lem3  26506  poimirlem26  32605  lkrpssN  33468  lnatexN  34083  lhpexle2lem  34313  lhpexle3lem  34315  cdlemg47  35042  cdlemk54  35264  tendoinvcl  35411  lcdlkreqN  35929  mapdh8ab  36084  jm2.26lem3  36586  stoweidlem36  38929