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Mirrors > Home > MPE Home > Th. List > 3imp3i2an | Structured version Visualization version GIF version |
Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2022.) |
Ref | Expression |
---|---|
3imp3i2an.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
3imp3i2an.2 | ⊢ ((𝜑 ∧ 𝜒) → 𝜏) |
3imp3i2an.3 | ⊢ ((𝜃 ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
3imp3i2an | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3imp3i2an.1 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
2 | 3imp3i2an.2 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜏) | |
3 | 2 | 3adant2 1129 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜏) |
4 | 3imp3i2an.3 | . 2 ⊢ ((𝜃 ∧ 𝜏) → 𝜂) | |
5 | 1, 3, 4 | syl2anc 583 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 |
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 396 df-3an 1087 |
This theorem is referenced by: focofo 6818 ordunel 7822 naddel1 8699 distrlem5pr 11036 divmul 11891 modmulnn 13872 moddi 13922 repswpfx 14753 shftval2 15040 pcgcd 16832 gsumccat 18778 qussub 19130 gsumdixp 20237 lspun 20853 evlslem4 21998 ordtcld3 23077 sleadd1im 27878 fusgrfisstep 29116 cplgr3v 29222 upgr2pthnlp 29520 frgrreg 30178 eliuniin 44378 eliuniin2 44399 disjinfi 44478 |
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