Theorem necon3ai 2557

Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon3ai.1	⊢ (φ → A = B)

Assertion

Ref	Expression
necon3ai	⊢ (A ≠ B → ¬ φ)

Proof of Theorem necon3ai

Step	Hyp	Ref	Expression
1		necon3ai.1	. . 3 ⊢ (φ → A = B)
2		nne 2521	. . 3 ⊢ (¬ A ≠ B ↔ A = B)
3	1, 2	sylibr 203	. 2 ⊢ (φ → ¬ A ≠ B)
4	3	con2i 112	1 ⊢ (A ≠ B → ¬ φ)

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

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by:  disjsn2  3788  fvunsn  5445  enadjlem1  6060  enadj  6061