Theorem neeq2 2526

Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)

Assertion

Ref	Expression
neeq2	⊢ (A = B → (C ≠ A ↔ C ≠ B))

Proof of Theorem neeq2

Step	Hyp	Ref	Expression
1		eqeq2 2362	. . 3 ⊢ (A = B → (C = A ↔ C = B))
2	1	notbid 285	. 2 ⊢ (A = B → (¬ C = A ↔ ¬ C = B))
3		df-ne 2519	. 2 ⊢ (C ≠ A ↔ ¬ C = A)
4		df-ne 2519	. 2 ⊢ (C ≠ B ↔ ¬ C = B)
5	2, 3, 4	3bitr4g 279	1 ⊢ (A = B → (C ≠ A ↔ C ≠ B))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2519

This theorem is used by:  neeq2i  2528  neeq2d  2531  psseq2  3358  nfunv  5139  enprmapc  6084  ce2  6193