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Mirrors > Home > MPE Home > Th. List > jaoian | Structured version Visualization version GIF version |
Description: Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.) |
Ref | Expression |
---|---|
jaoian.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
jaoian.2 | ⊢ ((𝜃 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
jaoian | ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jaoian.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 412 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | jaoian.2 | . . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒) | |
4 | 3 | ex 412 | . . 3 ⊢ (𝜃 → (𝜓 → 𝜒)) |
5 | 2, 4 | jaoi 857 | . 2 ⊢ ((𝜑 ∨ 𝜃) → (𝜓 → 𝜒)) |
6 | 5 | imp 406 | 1 ⊢ (((𝜑 ∨ 𝜃) ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 |
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 207 df-an 396 df-or 848 |
This theorem is referenced by: ccase 1037 preq12nebg 4868 opthprneg 4870 elpreqpr 4872 tpres 7221 xaddnemnf 13275 xaddnepnf 13276 faclbnd 14326 faclbnd3 14328 faclbnd4lem1 14329 znf1o 21588 degltlem1 26126 ipasslem3 30862 padct 32737 fz1nntr 32812 xrge0iifhom 33898 bj-ideqg1ALT 37148 nn0addcom 42457 nn0mulcom 42461 fzsplit1nn0 42742 f1mo 48683 |
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