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Mirrors > Home > MPE Home > Th. List > eupick | Structured version Visualization version GIF version |
Description: Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing 𝑥 such that 𝜑 is true, and there is also an 𝑥 (actually the same one) such that 𝜑 and 𝜓 are both true, then 𝜑 implies 𝜓 regardless of 𝑥. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.) |
Ref | Expression |
---|---|
eupick | ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2625 | . 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | mopick 2682 | . 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
3 | 1, 2 | sylan 580 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1765 ∃*wmo 2576 ∃!weu 2613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-10 2114 ax-12 2143 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 |
This theorem is referenced by: eupicka 2691 eupickb 2692 reupick 4213 reupick3 4214 eusv2nf 5194 reusv2lem3 5199 copsexg 5280 funssres 6275 oprabid 7054 txcn 21922 isch3 28705 bnj849 31809 iotasbc 40310 |
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