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Mirrors > Home > ILE Home > Th. List > eupick | 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 2070 | . 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | mopick 2116 | . 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
3 | 1, 2 | sylan 283 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∃wex 1503 ∃!weu 2038 ∃*wmo 2039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 |
This theorem is referenced by: eupicka 2118 eupickb 2119 reupick 3434 reupick3 3435 copsexg 4262 eusv2nf 4474 funssres 5277 oprabid 5927 txcn 14227 |
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