Theorem neeq1 2233

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 2062	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
2	1	notbid 602	. 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶))
3		df-ne 2221	. 2 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
4		df-ne 2221	. 2 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶)
5	2, 3, 4	3bitr4g 216	1 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 102   = wceq 1259   ≠ wne 2220

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038

This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-ne 2221

This theorem is referenced by:  neeq1i  2235  neeq1d  2238  nelrdva  2768  psseq1  3058  0inp0  3946  uzn0  8583  xrnemnf  8799  xrnepnf  8800  ngtmnft  8831  fztpval  9046