Theorem necon3d 2555

Description: Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)

Hypothesis

Ref	Expression
necon3d.1	⊢ (φ → (A = B → C = D))

Assertion

Ref	Expression
necon3d	⊢ (φ → (C ≠ D → A ≠ B))

Proof of Theorem necon3d

Step	Hyp	Ref	Expression
1		necon3d.1	. . 3 ⊢ (φ → (A = B → C = D))
2	1	necon3ad 2553	. 2 ⊢ (φ → (C ≠ D → ¬ A = B))
3		df-ne 2519	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
4	2, 3	syl6ibr 218	1 ⊢ (φ → (C ≠ D → A ≠ B))

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:  necon3i  2556  pm13.18  2589  ssn0  3584  pssdifn0  3612  uniintsn  3964  evenodddisj  4517  sfinltfin  4536  leltctr  6213  nnltp1c  6263