Theorem necon3bd 2554

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon3bd.1	⊢ (φ → (A = B → ψ))

Assertion

Ref	Expression
necon3bd	⊢ (φ → (¬ ψ → A ≠ B))

Proof of Theorem necon3bd

Step	Hyp	Ref	Expression
1		nne 2521	. . 3 ⊢ (¬ A ≠ B ↔ A = B)
2		necon3bd.1	. . 3 ⊢ (φ → (A = B → ψ))
3	1, 2	syl5bi 208	. 2 ⊢ (φ → (¬ A ≠ B → ψ))
4	3	con1d 116	1 ⊢ (φ → (¬ ψ → 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:  nelne1  2606  nelne2  2607  nssne1  3328  nssne2  3329  disjne  3597  difsn  3846  nbrne1  4657  nbrne2  4658