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Mirrors > Home > ILE Home > Th. List > mooran1 | GIF version |
Description: "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
mooran1 | ⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | moimi 2091 | . 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜑 ∧ 𝜓)) |
3 | moan 2095 | . 2 ⊢ (∃*𝑥𝜓 → ∃*𝑥(𝜑 ∧ 𝜓)) | |
4 | 2, 3 | jaoi 716 | 1 ⊢ ((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 ∃*wmo 2027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 |
This theorem is referenced by: (None) |
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