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Mirrors > Home > ILE Home > Th. List > con1bdc | GIF version |
Description: Contraposition. Bidirectional version of con1dc 856. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
con1bdc | ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con1dc 856 | . . . 4 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))) | |
2 | 1 | adantr 276 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))) |
3 | con1dc 856 | . . . 4 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜓))) | |
4 | 3 | adantl 277 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜓 → 𝜑) → (¬ 𝜑 → 𝜓))) |
5 | 2, 4 | impbid 129 | . 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑))) |
6 | 5 | ex 115 | 1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → 𝜑)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 834 |
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 614 ax-in2 615 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 |
This theorem is referenced by: (None) |
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