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Mirrors > Home > ILE Home > Th. List > con2biddc | GIF version |
Description: A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.) |
Ref | Expression |
---|---|
con2biddc.1 | ⊢ (𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒))) |
Ref | Expression |
---|---|
con2biddc | ⊢ (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con2biddc.1 | . . . 4 ⊢ (𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒))) | |
2 | bicom 140 | . . . 4 ⊢ ((𝜓 ↔ ¬ 𝜒) ↔ (¬ 𝜒 ↔ 𝜓)) | |
3 | 1, 2 | imbitrdi 161 | . . 3 ⊢ (𝜑 → (DECID 𝜒 → (¬ 𝜒 ↔ 𝜓))) |
4 | 3 | con1biddc 877 | . 2 ⊢ (𝜑 → (DECID 𝜒 → (¬ 𝜓 ↔ 𝜒))) |
5 | bicom 140 | . 2 ⊢ ((¬ 𝜓 ↔ 𝜒) ↔ (𝜒 ↔ ¬ 𝜓)) | |
6 | 4, 5 | imbitrdi 161 | 1 ⊢ (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ 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: anordc 957 xor3dc 1397 |
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