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Mirrors > Home > ILE Home > Th. List > 3anim123i | GIF version |
Description: Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.) |
Ref | Expression |
---|---|
3anim123i.1 | ⊢ (𝜑 → 𝜓) |
3anim123i.2 | ⊢ (𝜒 → 𝜃) |
3anim123i.3 | ⊢ (𝜏 → 𝜂) |
Ref | Expression |
---|---|
3anim123i | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anim123i.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 1 | 3ad2ant1 1003 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜓) |
3 | 3anim123i.2 | . . 3 ⊢ (𝜒 → 𝜃) | |
4 | 3 | 3ad2ant2 1004 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜃) |
5 | 3anim123i.3 | . . 3 ⊢ (𝜏 → 𝜂) | |
6 | 5 | 3ad2ant3 1005 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) |
7 | 2, 4, 6 | 3jca 1162 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 965 |
This theorem is referenced by: 3anim1i 1168 3anim2i 1169 3anim3i 1170 syl3an 1259 syl3anl 1268 spc3egv 2781 spc3gv 2782 eloprabga 5866 le2tri3i 7896 fzmmmeqm 9869 elfz1b 9901 elfz0fzfz0 9934 elfzmlbp 9940 elfzo1 9998 flltdivnn0lt 10108 modmulconst 11561 nndvdslegcd 11690 |
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