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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bian1d | Structured version Visualization version GIF version |
Description: Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.) |
Ref | Expression |
---|---|
bian1d.1 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
Ref | Expression |
---|---|
bian1d | ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bian1d.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | |
2 | 1 | biimpd 218 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) |
3 | 2 | adantld 482 | . 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) → (𝜒 ∧ 𝜃))) |
4 | simpl 472 | . . . 4 ⊢ ((𝜒 ∧ 𝜃) → 𝜒) | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜒)) |
6 | 1 | biimprd 237 | . . 3 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜓)) |
7 | 5, 6 | jcad 554 | . 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → (𝜒 ∧ 𝜓))) |
8 | 3, 7 | impbid 201 | 1 ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 |
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 196 df-an 385 |
This theorem is referenced by: funcnvmptOLD 28850 funcnvmpt 28851 |
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