Theorem neeq1 2348

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 2172	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2	1	notbid 657	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶))
3		df-ne 2336	. 2 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
4		df-ne 2336	. 2 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶)
5	2, 3, 4	3bitr4g 222	1 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1343   ≠ wne 2335

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 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-ne 2336

This theorem is referenced by:  neeq1i  2350  neeq1d  2353  nelrdva  2932  disji2  3974  0inp0  4144  frecabcl  6363  fiintim  6890  eldju2ndl  7033  updjudhf  7040  xnn0nemnf  9184  uzn0  9477  xrnemnf  9709  xrnepnf  9710  ngtmnft  9749  xsubge0  9813  xposdif  9814  xleaddadd  9819  fztpval  10014  pcpre1  12220  pcqmul  12231  pcqcl  12234  fiinopn  12602  neapmkv  13906