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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 608 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏))) |
4 | 3anim123d.3 | . . 3 ⊢ (𝜑 → (𝜂 → 𝜁)) | |
5 | 3, 4 | anim12d 608 | . 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜃) ∧ 𝜂) → ((𝜒 ∧ 𝜏) ∧ 𝜁))) |
6 | df-3an 1082 | . 2 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ ((𝜓 ∧ 𝜃) ∧ 𝜂)) | |
7 | df-3an 1082 | . 2 ⊢ ((𝜒 ∧ 𝜏 ∧ 𝜁) ↔ ((𝜒 ∧ 𝜏) ∧ 𝜁)) | |
8 | 5, 6, 7 | 3imtr4g 297 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) → (𝜒 ∧ 𝜏 ∧ 𝜁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 |
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 208 df-an 397 df-3an 1082 |
This theorem is referenced by: pofun 5386 isopolem 6968 issmo2 7845 smores 7848 inawina 9965 gchina 9974 repswcshw 14014 coprmprod 15838 issubmnd 17761 issubg2 18052 issubrg2 19249 ocv2ss 20503 sslm 21595 cmetcaulem 23578 axcontlem4 26440 axcontlem8 26444 redwlk 27140 clwwlknwwlksn 27502 clwwlknonwwlknonb 27571 numclwwlk1lem2foa 27821 dipsubdir 28312 subgrpth 31991 cgr3tr4 33124 idinside 33156 ftc1anclem7 34525 fzmul 34569 fdc1 34574 rngosubdi 34776 rngosubdir 34777 cdlemg33a 37394 upwlkwlk 43518 lidlmsgrp 43697 lidlrng 43698 |
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