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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 2636 | . 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | mopick 2673 | . 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
3 | 1, 2 | sylan 489 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1853 ∃!weu 2607 ∃*wmo 2608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-10 2168 ax-12 2196 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ex 1854 df-nf 1859 df-eu 2611 df-mo 2612 |
This theorem is referenced by: eupicka 2675 eupickb 2676 reupick 4054 reupick3 4055 eusv2nf 5013 reusv2lem3 5020 copsexg 5104 funssres 6091 oprabid 6841 txcn 21651 isch3 28428 bnj849 31323 iotasbc 39140 |
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