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Mirrors > Home > ILE Home > Th. List > necon4abiddc | GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.) |
Ref | Expression |
---|---|
necon4abiddc.1 | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓)))) |
Ref | Expression |
---|---|
necon4abiddc | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ 𝜓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon4abiddc.1 | . . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓)))) | |
2 | df-ne 2348 | . . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 2 | bibi1i 228 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ↔ ¬ 𝜓) ↔ (¬ 𝐴 = 𝐵 ↔ ¬ 𝜓)) |
4 | 1, 3 | syl8ib 166 | . 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝜓)))) |
5 | 4 | con4biddc 857 | 1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ 𝜓)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 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: necon4bbiddc 2421 necon4biddc 2422 lgsprme0 14314 |
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