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Mirrors > Home > MPE Home > Th. List > bi2bian9 | Structured version Visualization version GIF version |
Description: Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.) |
Ref | Expression |
---|---|
bi2an9.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
bi2an9.2 | ⊢ (𝜃 → (𝜏 ↔ 𝜂)) |
Ref | Expression |
---|---|
bi2bian9 | ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ↔ 𝜏) ↔ (𝜒 ↔ 𝜂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi2an9.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒)) |
3 | bi2an9.2 | . . 3 ⊢ (𝜃 → (𝜏 ↔ 𝜂)) | |
4 | 3 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 ↔ 𝜂)) |
5 | 2, 4 | bibi12d 346 | 1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ↔ 𝜏) ↔ (𝜒 ↔ 𝜂))) |
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: releccnveq 36397 extssr 36627 wepwsolem 40867 aomclem8 40886 |
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