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Mirrors > Home > ILE Home > Th. List > necon4biddc | GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.) |
Ref | Expression |
---|---|
necon4biddc.1 | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)))) |
Ref | Expression |
---|---|
necon4biddc | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon4biddc.1 | . . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)))) | |
2 | df-ne 2328 | . . . 4 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷) | |
3 | 2 | bibi2i 226 | . . 3 ⊢ ((𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐷)) |
4 | 1, 3 | syl8ib 165 | . 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐷)))) |
5 | 4 | necon4abiddc 2400 | 1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 DECID wdc 820 = wceq 1335 ≠ wne 2327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-ne 2328 |
This theorem is referenced by: nebidc 2407 |
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