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  6299  suppfnss  6497  suppssfvg  6503  nnmord  6790  findcard2  7193  findcard2s  7194  addn0nid  8700  nn0n0n1ge2  9717  xnegdi  10272  efne0  12447  divgcdcoprmex  12882  pceulem  13075  pcqmul  13084  pcqcl  13087  pcaddlem  13120  pcadd  13121  grpinvnz  13878  ringelnzr  14496  lmodfopne  14665  lmodindp1  14767  birthdaylem1g  16093  clwwlkccat  16654  clwwlknonel  16685