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Mirrors > Home > MPE Home > Th. List > Mathboxes > logic2 | Structured version Visualization version GIF version |
Description: Variant of logic1 45854. (Contributed by Zhi Wang, 30-Aug-2024.) |
Ref | Expression |
---|---|
pm4.71da.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
logic2.2 | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))) |
Ref | Expression |
---|---|
logic2 | ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.71da.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | pm4.71da 45853 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒))) |
3 | logic2.2 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏))) | |
4 | 2, 3 | sylbid 243 | . 2 ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜏))) |
5 | 1, 4 | logic1 45854 | 1 ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 |
This theorem is referenced by: ralbidc 45864 |
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