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Mirrors > Home > MPE Home > Th. List > 3anim123d | Structured version Visualization version GIF version |
Description: Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
3anim123d.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
3anim123d.2 | ⊢ (𝜑 → (𝜃 → 𝜏)) |
3anim123d.3 | ⊢ (𝜑 → (𝜂 → 𝜁)) |
Ref | Expression |
---|---|
3anim123d | ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) → (𝜒 ∧ 𝜏 ∧ 𝜁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anim123d.1 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 3anim123d.2 | . . . 4 ⊢ (𝜑 → (𝜃 → 𝜏)) | |
3 | 1, 2 | anim12d 609 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏))) |
4 | 3anim123d.3 | . . 3 ⊢ (𝜑 → (𝜂 → 𝜁)) | |
5 | 3, 4 | anim12d 609 | . 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜃) ∧ 𝜂) → ((𝜒 ∧ 𝜏) ∧ 𝜁))) |
6 | df-3an 1089 | . 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜂)) | |
7 | df-3an 1089 | . 2 ⊢ ((𝜒 ∧ 𝜏 ∧ 𝜁) ↔ ((𝜒 ∧ 𝜏) ∧ 𝜁)) | |
8 | 5, 6, 7 | 3imtr4g 295 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) → (𝜒 ∧ 𝜏 ∧ 𝜁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 |
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 df-3an 1089 |
This theorem is referenced by: pofun 5606 isopolem 7344 issmo2 8351 smores 8354 inawina 10687 gchina 10696 repswcshw 14764 coprmprod 16600 issubmnd 18654 issubg2 19023 issubrg2 20343 ocv2ss 21232 issubassa3 21426 sslm 22810 cmetcaulem 24812 axcontlem4 28263 axcontlem8 28267 redwlk 28967 clwwlknwwlksn 29329 numclwwlk1lem2foa 29645 dipsubdir 30139 subgrpth 34194 cgr3tr4 35093 idinside 35125 ftc1anclem7 36653 fzmul 36695 fdc1 36700 rngosubdi 36899 rngosubdir 36900 cdlemg33a 39663 upwlkwlk 46596 issubrng2 46816 rnglidlmsgrp 46836 rnglidlrng 46837 |
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