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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 128 | . . . 4 ⊢ ((ψ ↔ ¬ χ) ↔ (¬ χ ↔ ψ)) | |
3 | 1, 2 | syl6ib 150 | . . 3 ⊢ (φ → (DECID χ → (¬ χ ↔ ψ))) |
4 | 3 | con1biddc 769 | . 2 ⊢ (φ → (DECID χ → (¬ ψ ↔ χ))) |
5 | bicom 128 | . 2 ⊢ ((¬ ψ ↔ χ) ↔ (χ ↔ ¬ ψ)) | |
6 | 4, 5 | syl6ib 150 | 1 ⊢ (φ → (DECID χ → (χ ↔ ¬ ψ))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 DECID wdc 741 |
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 629 |
This theorem depends on definitions: df-bi 110 df-dc 742 |
This theorem is referenced by: anordc 862 xor3dc 1275 |
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