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Mirrors > Home > ILE Home > Th. List > necon1addc | GIF version |
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.) |
Ref | Expression |
---|---|
necon1addc.1 | ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 = 𝐵))) |
Ref | Expression |
---|---|
necon1addc | ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2283 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon1addc.1 | . . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 = 𝐵))) | |
3 | con1dc 824 | . . 3 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝐴 = 𝐵) → (¬ 𝐴 = 𝐵 → 𝜓))) | |
4 | 2, 3 | sylcom 28 | . 2 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝐴 = 𝐵 → 𝜓))) |
5 | 1, 4 | syl7bi 164 | 1 ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 802 = wceq 1314 ≠ wne 2282 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 |
This theorem depends on definitions: df-bi 116 df-stab 799 df-dc 803 df-ne 2283 |
This theorem is referenced by: (None) |
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