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Mirrors > Home > MPE Home > Th. List > eupickb | Structured version Visualization version GIF version |
Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Ref | Expression |
---|---|
eupickb | ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupick 2633 | . . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) | |
2 | 1 | 3adant2 1130 | . 2 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓)) |
3 | exancom 1863 | . . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) | |
4 | eupick 2633 | . . . 4 ⊢ ((∃!𝑥𝜓 ∧ ∃𝑥(𝜓 ∧ 𝜑)) → (𝜓 → 𝜑)) | |
5 | 3, 4 | sylan2b 594 | . . 3 ⊢ ((∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑)) |
6 | 5 | 3adant1 1129 | . 2 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑)) |
7 | 2, 6 | impbid 211 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 ∃wex 1780 ∃!weu 2566 |
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 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-ex 1781 df-nf 1785 df-mo 2538 df-eu 2567 |
This theorem is referenced by: (None) |
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