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Mirrors > Home > ILE Home > Th. List > necon3ad | GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.) |
Ref | Expression |
---|---|
necon3ad.1 | ⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵)) |
Ref | Expression |
---|---|
necon3ad | ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2257 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon3ad.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵)) | |
3 | 2 | con3d 597 | . 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝜓)) |
4 | 1, 3 | syl5bi 151 | 1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1290 ≠ wne 2256 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-in1 580 ax-in2 581 |
This theorem depends on definitions: df-bi 116 df-ne 2257 |
This theorem is referenced by: necon3d 2300 disjnim 3842 nlt1pig 6961 eucalglt 11378 nprm 11444 |
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