Theorem necon3d 2371

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 2369	. 2 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵))
3		df-ne 2328	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	syl6ibr 161	1 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1335   ≠ wne 2327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2328

This theorem is referenced by:  necon3i  2375  pm13.18  2408  ssn0  3436  suppssfv  6025  suppssov1  6026  nnmord  6461  findcard2  6831  findcard2s  6832  addn0nid  8243  nn0n0n1ge2  9228  xnegdi  9765  efne0  11568  divgcdcoprmex  11970