Theorem necon3d 2464

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 2462	. 2 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵))
3		df-ne 2421	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	imbitrrdi 162	1 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1402   ≠ wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  necon3i  2468  pm13.18  2501  ssn0  3566  suppssov1  6302  suppfnss  6501  suppssfvg  6507  nnmord  6794  findcard2  7198  findcard2s  7199  addn0nid  8708  nn0n0n1ge2  9727  xnegdi  10288  efne0  12471  divgcdcoprmex  12906  pceulem  13103  pcqmul  13112  pcqcl  13115  pcaddlem  13148  pcadd  13149  grpinvnz  13936  ringelnzr  14604  lmodfopne  14773  lmodindp1  14875  birthdaylem1g  16237  clwwlkccat  16859  clwwlknonel  16890