Theorem 3netr3d 3009

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 3002	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
5	1, 4	eqnetrrd 3001	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2933

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-ne 2934

This theorem is referenced by:  subrgnzr  20539  clmopfne  25064  dchrisum0re  27492  fracfld  33401  qsnzr  33547  dimlssid  33809  algextdeglem4  33897  constrrtll  33908  cdlemg9a  41002  cdlemg11aq  41008  cdlemg12b  41014  cdlemg12  41020  cdlemg13  41022  cdlemg19  41054  cdlemk3  41203  cdlemk12  41220  cdlemk12u  41242  lclkrlem2g  41883  mapdncol  42040  mapdpglem29  42070  hdmaprnlem1N  42219  hdmap14lem9  42246  aks6d1c2p2  42483  ricdrng1  42892  pellex  43186