Theorem neeq1 2413

Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Assertion

Ref	Expression
neeq1	⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))

Proof of Theorem neeq1

Step	Hyp	Ref	Expression
1		eqeq1 2236	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2	1	notbid 671	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶))
3		df-ne 2401	. 2 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
4		df-ne 2401	. 2 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶)
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1395   ≠ wne 2400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-ne 2401

This theorem is referenced by:  neeq1i  2415  neeq1d  2418  nelrdva  3010  disji2  4074  0inp0  4249  frecabcl  6543  fiintim  7089  eldju2ndl  7235  updjudhf  7242  netap  7436  2oneel  7438  2omotaplemap  7439  2omotaplemst  7440  exmidapne  7442  xnn0nemnf  9439  uzn0  9734  xrnemnf  9969  xrnepnf  9970  ngtmnft  10009  xsubge0  10073  xposdif  10074  xleaddadd  10079  fztpval  10275  hashdmprop2dom  11061  fun2dmnop0  11064  pcpre1  12810  pcqmul  12821  pcqcl  12824  xpsfrnel  13372  isnzr2  14142  fiinopn  14672  umgrvad2edg  16003  neapmkv  16395  neap0mkv  16396  ltlenmkv  16397