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Mirrors > Home > ILE Home > Th. List > con2bidc | GIF version |
Description: Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.) |
Ref | Expression |
---|---|
con2bidc | ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con1bidc 875 | . . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑)))) | |
2 | 1 | imp 124 | . . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ↔ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑))) |
3 | bicom 140 | . . . 4 ⊢ ((¬ 𝜑 ↔ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)) | |
4 | bicom 140 | . . . 4 ⊢ ((¬ 𝜓 ↔ 𝜑) ↔ (𝜑 ↔ ¬ 𝜓)) | |
5 | 2, 3, 4 | 3bitr3g 222 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜓 ↔ ¬ 𝜑) ↔ (𝜑 ↔ ¬ 𝜓))) |
6 | 5 | bicomd 141 | . 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))) |
7 | 6 | ex 115 | 1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 |
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 615 ax-in2 616 ax-io 710 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 |
This theorem is referenced by: annimdc 938 pm4.55dc 939 orandc 940 nbbndc 1404 |
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