Theorem necon3d 2391

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 2389	. 2 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵))
3		df-ne 2348	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	imbitrrdi 162	1 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1353   ≠ wne 2347

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 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-ne 2348

This theorem is referenced by:  necon3i  2395  pm13.18  2428  ssn0  3467  suppssfv  6082  suppssov1  6083  nnmord  6521  findcard2  6892  findcard2s  6893  addn0nid  8334  nn0n0n1ge2  9326  xnegdi  9871  efne0  11689  divgcdcoprmex  12105  pceulem  12297  pcqmul  12306  pcqcl  12309  pcaddlem  12341  pcadd  12342  grpinvnz  12947  ringelnzr  13334  lmodfopne  13422  lmodindp1  13520