Theorem 3netr4d 3010

Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 21-Nov-2019.)

Hypotheses

Ref	Expression
3netr4d.1	⊢ (𝜑 → 𝐴 ≠ 𝐵)
3netr4d.2	⊢ (𝜑 → 𝐶 = 𝐴)
3netr4d.3	⊢ (𝜑 → 𝐷 = 𝐵)

Assertion

Ref	Expression
3netr4d	⊢ (𝜑 → 𝐶 ≠ 𝐷)

Proof of Theorem 3netr4d

Step	Hyp	Ref	Expression
1		3netr4d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐴)
2		3netr4d.1	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
3	1, 2	eqnetrd 3000	. 2 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
4		3netr4d.3	. 2 ⊢ (𝜑 → 𝐷 = 𝐵)
5	3, 4	neeqtrrd 3007	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:  f1ounsn  7220  infpssrlem4  10219  modsumfzodifsn  13897  mgm2nsgrplem4  18883  pmtr3ncomlem1  19439  isdrng2  20711  prmirredlem  21462  uvcf1  21782  dfac14lem  23592  i1fmullem  25671  fta1glem1  26143  fta1blem  26146  plydivlem4  26273  fta1lem  26284  cubic  26826  asinlem  26845  dchrn0  27227  lgsne0  27312  noextenddif  27646  noresle  27675  perpneq  28796  axlowdimlem14  29038  preimane  32757  cycpmco2lem6  33207  cycpmrn  33219  mplmulmvr  33698  irngnminplynz  33872  cntnevol  34388  subfacp1lem5  35382  fvtransport  36230  mh-inf3f1  36739  poimirlem1  37956  poimirlem6  37961  poimirlem7  37962  dalem4  40125  cdleme35sn2aw  40918  cdleme39n  40926  cdleme41fva11  40937  trlcone  41188  hdmaprnlem3N  42310  sticksstones2  42600  expeq1d  42770  uvcn0  43001  flt0  43084  stoweidlem23  46469  gpg3kgrtriex  48577  gpgprismgr4cycllem7  48589  2zrngnmlid  48743  2zrngnmrid  48744  zlmodzxznm  48985  line2y  49243