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Mirrors > Home > ILE Home > Th. List > necon1bddc | GIF version |
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.) |
Ref | Expression |
---|---|
necon1bddc.1 | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜓))) |
Ref | Expression |
---|---|
necon1bddc | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 → 𝐴 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon1bddc.1 | . . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜓))) | |
2 | df-ne 2263 | . . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 2 | imbi1i 237 | . . 3 ⊢ ((𝐴 ≠ 𝐵 → 𝜓) ↔ (¬ 𝐴 = 𝐵 → 𝜓)) |
4 | 1, 3 | syl6ib 160 | . 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → 𝜓))) |
5 | con1dc 794 | . 2 ⊢ (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵 → 𝜓) → (¬ 𝜓 → 𝐴 = 𝐵))) | |
6 | 4, 5 | sylcom 28 | 1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 → 𝐴 = 𝐵))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 783 = wceq 1296 ≠ wne 2262 |
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 582 ax-in2 583 ax-io 668 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-ne 2263 |
This theorem is referenced by: necon1ddc 2340 |
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