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Mirrors > Home > ILE Home > Th. List > con1biidc | GIF version |
Description: A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.) |
Ref | Expression |
---|---|
con1biidc.1 | ⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
con1biidc | ⊢ (DECID 𝜑 → (¬ 𝜓 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotbdc 810 | . . 3 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ ¬ 𝜑)) | |
2 | con1biidc.1 | . . . 4 ⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝜓)) | |
3 | 2 | notbid 633 | . . 3 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 ↔ ¬ 𝜓)) |
4 | 1, 3 | bitrd 187 | . 2 ⊢ (DECID 𝜑 → (𝜑 ↔ ¬ 𝜓)) |
5 | 4 | bicomd 140 | 1 ⊢ (DECID 𝜑 → (¬ 𝜓 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 DECID wdc 786 |
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 584 ax-in2 585 ax-io 671 |
This theorem depends on definitions: df-bi 116 df-dc 787 |
This theorem is referenced by: con2biidc 817 necon1abiidc 2327 necon1bbiidc 2328 |
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