Theorem 3netr3d 3008

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 3001	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
5	1, 4	eqnetrrd 3000	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2728  df-ne 2933

This theorem is referenced by:  subrgnzr  20571  clmopfne  25063  dchrisum0re  27476  fracfld  33369  qsnzr  33515  dimlssid  33776  algextdeglem4  33864  constrrtll  33875  cdlemg9a  41078  cdlemg11aq  41084  cdlemg12b  41090  cdlemg12  41096  cdlemg13  41098  cdlemg19  41130  cdlemk3  41279  cdlemk12  41296  cdlemk12u  41318  lclkrlem2g  41959  mapdncol  42116  mapdpglem29  42146  hdmaprnlem1N  42295  hdmap14lem9  42322  aks6d1c2p2  42558  ricdrng1  42973  pellex  43263