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  2968  disji2  4023  0inp0  4196  frecabcl  6454  fiintim  6987  eldju2ndl  7133  updjudhf  7140  netap  7316  2oneel  7318  2omotaplemap  7319  2omotaplemst  7320  exmidapne  7322  xnn0nemnf  9317  uzn0  9611  xrnemnf  9846  xrnepnf  9847  ngtmnft  9886  xsubge0  9950  xposdif  9951  xleaddadd  9956  fztpval  10152  pcpre1  12433  pcqmul  12444  pcqcl  12447  xpsfrnel  12930  isnzr2  13683  fiinopn  14183  neapmkv  15628  neap0mkv  15629  ltlenmkv  15630