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Mirrors > Home > MPE Home > Th. List > mopick2 | Structured version Visualization version GIF version |
Description: "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1890. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
mopick2 | ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmo1 2552 | . . . 4 ⊢ Ⅎ𝑥∃*𝑥𝜑 | |
2 | nfe1 2148 | . . . 4 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ 𝜓) | |
3 | 1, 2 | nfan 1903 | . . 3 ⊢ Ⅎ𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) |
4 | mopick 2622 | . . . . . 6 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
5 | 4 | ancld 552 | . . . . 5 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → (𝜑 ∧ 𝜓))) |
6 | 5 | anim1d 612 | . . . 4 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ((𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∧ 𝜒))) |
7 | df-3an 1090 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
8 | 6, 7 | imbitrrdi 251 | . . 3 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ((𝜑 ∧ 𝜒) → (𝜑 ∧ 𝜓 ∧ 𝜒))) |
9 | 3, 8 | eximd 2210 | . 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (∃𝑥(𝜑 ∧ 𝜒) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))) |
10 | 9 | 3impia 1118 | 1 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∃wex 1782 ∃*wmo 2533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-mo 2535 |
This theorem is referenced by: moantr 37233 |
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