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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 2576 | . 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | mopick 2623 | . 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 395 ∃wex 1776 ∃*wmo 2536 ∃!weu 2566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-mo 2538 df-eu 2567 |
This theorem is referenced by: eupicka 2632 eupickb 2633 reupick 4335 reupick3 4336 eusv2nf 5401 reusv2lem3 5406 copsexgw 5501 copsexg 5502 funssres 6612 oprabidw 7462 oprabid 7463 txcn 23650 isch3 31270 bnj849 34918 iotasbc 44415 |
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