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Mirrors > Home > ILE Home > Th. List > necon1ddc | GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.) |
Ref | Expression |
---|---|
necon1ddc.1 | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷))) |
Ref | Expression |
---|---|
necon1ddc | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2348 | . 2 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷) | |
2 | necon1ddc.1 | . . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷))) | |
3 | 2 | necon1bddc 2424 | . 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐶 = 𝐷 → 𝐴 = 𝐵))) |
4 | 1, 3 | syl7bi 165 | 1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 834 = wceq 1353 ≠ wne 2347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-ne 2348 |
This theorem is referenced by: xblss2ps 13537 xblss2 13538 lgsne0 14072 |
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