Theorem necon3d 2458

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 2456	. 2 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵))
3		df-ne 2415	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	imbitrrdi 162	1 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1398   ≠ wne 2414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-ne 2415

This theorem is referenced by:  necon3i  2462  pm13.18  2495  ssn0  3553  suppssov1  6265  suppfnss  6459  suppssfvg  6465  nnmord  6752  findcard2  7148  findcard2s  7149  addn0nid  8652  nn0n0n1ge2  9653  xnegdi  10207  efne0  12372  divgcdcoprmex  12807  pceulem  13000  pcqmul  13009  pcqcl  13012  pcaddlem  13045  pcadd  13046  grpinvnz  13805  ringelnzr  14354  lmodfopne  14523  lmodindp1  14625  clwwlkccat  16445  clwwlknonel  16476