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Mirrors > Home > ILE Home > Th. List > reupick3 | GIF version |
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.) |
Ref | Expression |
---|---|
reupick3 | ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 2421 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | df-rex 2420 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
3 | anass 398 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
4 | 3 | exbii 1584 | . . . . 5 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) |
5 | 2, 4 | bitr4i 186 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) |
6 | eupick 2076 | . . . 4 ⊢ ((∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) | |
7 | 1, 5, 6 | syl2anb 289 | . . 3 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)) |
8 | 7 | expd 256 | . 2 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) → (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) |
9 | 8 | 3impia 1178 | 1 ⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 ∃wex 1468 ∈ wcel 1480 ∃!weu 1997 ∃wrex 2415 ∃!wreu 2416 |
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-3an 964 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-rex 2420 df-reu 2421 |
This theorem is referenced by: reupick2 3357 |
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