Theorem neeq1 2380

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 2203	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2	1	notbid 668	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶))
3		df-ne 2368	. 2 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
4		df-ne 2368	. 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 2367

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-ne 2368

This theorem is referenced by:  neeq1i  2382  neeq1d  2385  nelrdva  2971  disji2  4027  0inp0  4200  frecabcl  6466  fiintim  7001  eldju2ndl  7147  updjudhf  7154  netap  7339  2oneel  7341  2omotaplemap  7342  2omotaplemst  7343  exmidapne  7345  xnn0nemnf  9342  uzn0  9636  xrnemnf  9871  xrnepnf  9872  ngtmnft  9911  xsubge0  9975  xposdif  9976  xleaddadd  9981  fztpval  10177  pcpre1  12488  pcqmul  12499  pcqcl  12502  xpsfrnel  13048  isnzr2  13818  fiinopn  14348  neapmkv  15825  neap0mkv  15826  ltlenmkv  15827