Theorem 3netr3d 3015

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 3008	. 2 ⊢ (𝜑 → 𝐴 ≠ 𝐷)
5	1, 4	eqnetrrd 3007	1 ⊢ (𝜑 → 𝐶 ≠ 𝐷)

Syntax hints:   → wi 4   = wceq 1537   ≠ wne 2938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ne 2939

This theorem is referenced by:  subrgnzr  20611  clmopfne  25143  dchrisum0re  27572  fracfld  33290  qsnzr  33463  dimlssid  33660  algextdeglem4  33726  constrrtll  33737  cdlemg9a  40615  cdlemg11aq  40621  cdlemg12b  40627  cdlemg12  40633  cdlemg13  40635  cdlemg19  40667  cdlemk3  40816  cdlemk12  40833  cdlemk12u  40855  lclkrlem2g  41496  mapdncol  41653  mapdpglem29  41683  hdmaprnlem1N  41832  hdmap14lem9  41859  aks6d1c2p2  42101  ricdrng1  42515  pellex  42823