Theorem 3netr3d 2858

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)

Hypotheses

Ref	Expression
3netr3d.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
3netr3d.2	⊢ (𝜑 → 𝐴 = 𝐶)
3netr3d.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
3netr3d	⊢ (𝜑 → 𝐶 ≠ 𝐷)

Proof of Theorem 3netr3d

Step	Hyp	Ref	Expression
1		3netr3d.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
2		3netr3d.1	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
3		3netr3d.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 3	neeqtrd 2851	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
5	1, 4	eqnetrrd 2850	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

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:  subrgnzr  19089  clmopfne  22704  dchrisum0re  25002  cdlemg9a  34938  cdlemg11aq  34944  cdlemg12b  34950  cdlemg12  34956  cdlemg13  34958  cdlemg19  34990  cdlemk3  35139  cdlemk12  35156  cdlemk12u  35178  lclkrlem2g  35820  mapdncol  35977  mapdpglem29  36007  hdmaprnlem1N  36159  hdmap14lem9  36186  pellex  36417