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Mirrors > Home > ILE Home > Th. List > con1biimdc | GIF version |
Description: Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.) |
Ref | Expression |
---|---|
con1biimdc | ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi1 117 | . . 3 ⊢ ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜑 → 𝜓)) | |
2 | con1dc 792 | . . 3 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))) | |
3 | 1, 2 | syl5 32 | . 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 → 𝜑))) |
4 | bi2 129 | . . . 4 ⊢ ((¬ 𝜑 ↔ 𝜓) → (𝜓 → ¬ 𝜑)) | |
5 | 4 | con2d 590 | . . 3 ⊢ ((¬ 𝜑 ↔ 𝜓) → (𝜑 → ¬ 𝜓)) |
6 | 5 | a1i 9 | . 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (𝜑 → ¬ 𝜓))) |
7 | 3, 6 | impbidd 126 | 1 ⊢ (DECID 𝜑 → ((¬ 𝜑 ↔ 𝜓) → (¬ 𝜓 ↔ 𝜑))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 DECID wdc 781 |
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 580 ax-in2 581 ax-io 666 |
This theorem depends on definitions: df-bi 116 df-dc 782 |
This theorem is referenced by: con1bidc 807 con1biddc 809 |
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