Theorem syl5eqner 2542

Description: B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)

Hypotheses

Ref	Expression
syl5eqner.1	⊢ B = A
syl5eqner.2	⊢ (φ → B ≠ C)

Assertion

Ref	Expression
syl5eqner	⊢ (φ → A ≠ C)

Proof of Theorem syl5eqner

Step	Hyp	Ref	Expression
1		syl5eqner.2	. 2 ⊢ (φ → B ≠ C)
2		syl5eqner.1	. . 3 ⊢ B = A
3	2	neeq1i 2527	. 2 ⊢ (B ≠ C ↔ A ≠ C)
4	1, 3	sylib 188	1 ⊢ (φ → A ≠ C)

Syntax hints:   → wi 4   = 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: (None)