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Theorem necon3d 2554

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 2552	. 2 ⊢ (φ → (C ≠ D → ¬ A = B))
3		df-ne 2518	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
4	2, 3	syl6ibr 218	1 ⊢ (φ → (C ≠ D → A ≠ B))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1642 ≠ wne 2516

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8

This theorem depends on definitions: df-bi 177 df-ne 2518

This theorem is referenced by: necon3i 2555 pm13.18 2588 ssn0 3583 pssdifn0 3611 uniintsn 3963 evenodddisj 4516 sfinltfin 4535 leltctr 6212 nnltp1c 6262