Theorem neeq1 2377

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 2200	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2	1	notbid 668	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶))
3		df-ne 2365	. 2 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
4		df-ne 2365	. 2 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶)
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   = wceq 1364   ≠ wne 2364

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 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-ne 2365

This theorem is referenced by:  neeq1i  2379  neeq1d  2382  nelrdva  2967  disji2  4022  0inp0  4195  frecabcl  6452  fiintim  6985  eldju2ndl  7131  updjudhf  7138  netap  7314  2oneel  7316  2omotaplemap  7317  2omotaplemst  7318  exmidapne  7320  xnn0nemnf  9314  uzn0  9608  xrnemnf  9843  xrnepnf  9844  ngtmnft  9883  xsubge0  9947  xposdif  9948  xleaddadd  9953  fztpval  10149  pcpre1  12430  pcqmul  12441  pcqcl  12444  xpsfrnel  12927  isnzr2  13680  fiinopn  14172  neapmkv  15558  neap0mkv  15559  ltlenmkv  15560