Theorem neeq1d 2530

Description: Deduction for inequality. (Contributed by NM, 25-Oct-1999.)

Hypothesis

Ref	Expression
neeq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
neeq1d	⊢ (φ → (A ≠ C ↔ B ≠ C))

Proof of Theorem neeq1d

Step	Hyp	Ref	Expression
1		neeq1d.1	. 2 ⊢ (φ → A = B)
2		neeq1 2525	. 2 ⊢ (A = B → (A ≠ C ↔ B ≠ C))
3	1, 2	syl 15	1 ⊢ (φ → (A ≠ C ↔ B ≠ C))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ≠ wne 2517

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346  df-ne 2519

This theorem is referenced by:  neeq12d  2532  eqnetrd  2535  prnzg  3837  preaddccan2lem1  4455  preaddccan2  4456  evenodddisj  4517  vfinncvntnn  4549  ereldm  5972  map0  6026  ce0addcnnul  6180  ce0nn  6181  ce0nnulb  6183