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Mirrors > Home > ILE Home > Th. List > necon3ai | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.) |
Ref | Expression |
---|---|
necon3ai.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
necon3ai | ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2206 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon3ai.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | con3i 562 | . 2 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝜑) |
4 | 1, 3 | sylbi 114 | 1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 ≠ wne 2204 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-in1 544 ax-in2 545 |
This theorem depends on definitions: df-bi 110 df-ne 2206 |
This theorem is referenced by: disjsn2 3433 0nelxp 4372 fvunsng 5357 |
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