Theorem neeq2 2414

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

Assertion

Ref	Expression
neeq2	⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))

Proof of Theorem neeq2

Step	Hyp	Ref	Expression
1		eqeq2 2239	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵))
2	1	notbid 671	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐶 = 𝐴 ↔ ¬ 𝐶 = 𝐵))
3		df-ne 2401	. 2 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴)
4		df-ne 2401	. 2 ⊢ (𝐶 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐵)
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))

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:  neeq2i  2416  neeq2d  2419  disji2  4078  fodjuomnilemdc  7334  netap  7463  2oneel  7465  2omotaplemap  7466  2omotaplemst  7467  exmidapne  7469  xrlttri3  10022  hashdmprop2dom  11098  fun2dmnop0  11101  isnzr2  14188  umgrvad2edg  16050  3dom  16523  neapmkv  16608  neap0mkv  16609  ltlenmkv  16610