Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > necon3d | GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.) |
Ref | Expression |
---|---|
necon3d.1 | ⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 = 𝐷)) |
Ref | Expression |
---|---|
necon3d | ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon3d.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 = 𝐷)) | |
2 | 1 | necon3ad 2350 | . 2 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → ¬ 𝐴 = 𝐵)) |
3 | df-ne 2309 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
4 | 2, 3 | syl6ibr 161 | 1 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 ≠ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ≠ wne 2308 |
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 603 ax-in2 604 |
This theorem depends on definitions: df-bi 116 df-ne 2309 |
This theorem is referenced by: necon3i 2356 pm13.18 2389 ssn0 3405 suppssfv 5978 suppssov1 5979 nnmord 6413 findcard2 6783 findcard2s 6784 addn0nid 8136 nn0n0n1ge2 9121 xnegdi 9651 efne0 11384 divgcdcoprmex 11783 |
Copyright terms: Public domain | W3C validator |