Theorem neeq12d 2843

Description: Deduction for inequality. (Contributed by NM, 24-Jul-2012.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Hypotheses

Ref	Expression
neeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
neeq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
neeq12d	⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷))

Proof of Theorem neeq12d

Step	Hyp	Ref	Expression
1		neeq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		neeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
3	1, 2	eqeq12d 2625	. 2 ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	3	necon3bid 2826	1 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷))

Syntax hints:   → wi 4   ↔ wb 195   = 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-an 385  df-cleq 2603  df-ne 2782

This theorem is referenced by:  fnelnfp  6348  suppval  7184  infpssrlem4  9011  injresinjlem  12450  sgrp2nmndlem5  17239  pmtr3ncom  17718  isnzr  19080  ptcmplem2  21667  isinag  25529  axlowdimlem6  25627  axlowdimlem14  25635  usgrcyclnl1  26168  constr3lem6  26177  4cycl4dv4e  26196  usg2wotspth  26411  2spontn0vne  26414  numclwwlk2lem1  26629  numclwlk2lem2f  26630  numclwlk2lem2f1o  26632  numclwwlk5  26639  signsvvfval  29981  signsvfn  29985  bnj1534  30177  bnj1542  30181  bnj1280  30342  derangsn  30406  derangenlem  30407  subfacp1lem3  30418  subfacp1lem5  30420  subfacp1lem6  30421  subfacp1  30422  sltval2  31053  sltres  31061  noseponlem  31065  nodenselem3  31082  nodenselem5  31084  nodenselem7  31086  nofulllem4  31104  nofulllem5  31105  fvtransport  31309  poimirlem1  32580  poimirlem6  32585  poimirlem7  32586  cdlemkid3N  35239  cdlemkid4  35240  stoweidlem43  38936  pthdadjvtx  40936  uhgr1wlkspthlem2  40960  usgr2wlkspth  40965  usgr2trlncl  40966  pthdlem1  40972  lfgrn1cycl  41008  21wlkdlem5  41136  2pthdlem1  41137  31wlkdlem5  41330  3pthdlem1  41331  av-numclwwlk2lem1  41532  av-numclwlk2lem2f  41533  av-numclwlk2lem2f1o  41535  av-numclwwlk5  41542  nnsgrpnmnd  41608  2zrngnmlid  41739  pgrpgt2nabl  41941  ldepsnlinc  42091