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Mirrors > Home > ILE Home > Th. List > con1biddc | GIF version |
Description: A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.) |
Ref | Expression |
---|---|
con1biddc.1 | ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
con1biddc | ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜒 ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con1biddc.1 | . 2 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ 𝜒))) | |
2 | con1biimdc 767 | . 2 ⊢ (DECID 𝜓 → ((¬ 𝜓 ↔ 𝜒) → (¬ 𝜒 ↔ 𝜓))) | |
3 | 1, 2 | sylcom 25 | 1 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜒 ↔ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 DECID wdc 742 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: con2biddc 774 pm5.18dc 777 necon1abiddc 2267 necon1bbiddc 2268 |
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