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Mirrors > Home > MPE Home > Th. List > anim12ii | Structured version Visualization version GIF version |
Description: Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.) |
Ref | Expression |
---|---|
anim12ii.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
anim12ii.2 | ⊢ (𝜃 → (𝜓 → 𝜏)) |
Ref | Expression |
---|---|
anim12ii | ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜒 ∧ 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anim12ii.1 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | anim12ii.2 | . 2 ⊢ (𝜃 → (𝜓 → 𝜏)) | |
3 | pm3.43 474 | . 2 ⊢ (((𝜓 → 𝜒) ∧ (𝜓 → 𝜏)) → (𝜓 → (𝜒 ∧ 𝜏))) | |
4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 → (𝜒 ∧ 𝜏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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: im2anan9 620 pm5.31r 829 2mo 2652 elex22 3453 disj 4387 tz7.2 5573 funcnvuni 7766 upgrwlkdvdelem 28092 funressnfv 44497 |
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