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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 2663 | . 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | mopick 2710 | . 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
3 | 1, 2 | sylan 582 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃wex 1780 ∃*wmo 2620 ∃!weu 2653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 |
This theorem is referenced by: eupicka 2719 eupickb 2720 reupick 4287 reupick3 4288 eusv2nf 5296 reusv2lem3 5301 copsexgw 5381 copsexg 5382 funssres 6398 oprabidw 7187 oprabid 7188 txcn 22234 isch3 29018 bnj849 32197 iotasbc 40771 |
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