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Mirrors > Home > ILE Home > Th. List > euim | GIF version |
Description: Add existential unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
Ref | Expression |
---|---|
euim | ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . 3 ⊢ (∃𝑥𝜑 → (∃!𝑥𝜓 → ∃𝑥𝜑)) | |
2 | euimmo 2064 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑)) | |
3 | 1, 2 | anim12ii 340 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → (∃𝑥𝜑 ∧ ∃*𝑥𝜑))) |
4 | eu5 2044 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | |
5 | 3, 4 | syl6ibr 161 | 1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 ∃wex 1468 ∃!weu 1997 ∃*wmo 1998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 |
This theorem is referenced by: (None) |
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