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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpbiran3d | Structured version Visualization version GIF version |
Description: Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024.) |
Ref | Expression |
---|---|
mpbiran3d.1 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
mpbiran3d.2 | ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
mpbiran3d | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpbiran3d.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | |
2 | 1 | simprbda 499 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
3 | 2 | ex 413 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) |
4 | mpbiran3d.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | |
5 | 4 | ex 413 | . . . 4 ⊢ (𝜑 → (𝜒 → 𝜃)) |
6 | 5 | ancld 551 | . . 3 ⊢ (𝜑 → (𝜒 → (𝜒 ∧ 𝜃))) |
7 | 6, 1 | sylibrd 258 | . 2 ⊢ (𝜑 → (𝜒 → 𝜓)) |
8 | 3, 7 | impbid 211 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 |
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 206 df-an 397 |
This theorem is referenced by: mpbiran4d 46121 functhinc 46304 thincsect 46316 thincinv 46318 grptcmon 46355 grptcepi 46356 |
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