Theorem neeq2 2341

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 2167	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵))
2	1	notbid 657	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐶 = 𝐴 ↔ ¬ 𝐶 = 𝐵))
3		df-ne 2328	. 2 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴)
4		df-ne 2328	. 2 ⊢ (𝐶 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐵)
5	2, 3, 4	3bitr4g 222	1 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1335   ≠ wne 2327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-ne 2328

This theorem is referenced by:  neeq2i  2343  neeq2d  2346  disji2  3958  fodjuomnilemdc  7082  xrlttri3  9699  neapmkv  13625