Theorem neeq2 2434

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 2248	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵))
2	1	notbid 677	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐶 = 𝐴 ↔ ¬ 𝐶 = 𝐵))
3		df-ne 2421	. 2 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴)
4		df-ne 2421	. 2 ⊢ (𝐶 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐵)
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1402   ≠ wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-ne 2421

This theorem is used by:  neeq2i  2436  neeq2d  2439  disji2  4122  fodjuomnilemdc  7484  netap  7620  2oneel  7622  2omotaplemap  7623  2omotaplemst  7624  exmidapne  7626  xrlttri3  10209  hashdmprop2dom  11310  fun2dmnop0  11316  isnzr2  14540  umgrvad2edg  16550  eupth2lem3lem4fi  16812  3dom  17116  qdiff  17196  neapmkv  17216  neap0mkv  17217  ltlenmkv  17218