Theorem neeq2 2296

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 2124	. . 3 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵))
2	1	notbid 639	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐶 = 𝐴 ↔ ¬ 𝐶 = 𝐵))
3		df-ne 2283	. 2 ⊢ (𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴)
4		df-ne 2283	. 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 1314   ≠ wne 2282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-4 1470  ax-17 1489  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-ne 2283

This theorem is referenced by:  neeq2i  2298  neeq2d  2301  disji2  3888  fodjuomnilemdc  6966  xrlttri3  9476