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Mirrors > Home > MPE Home > Th. List > euop2 | Structured version Visualization version GIF version |
Description: Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.) |
Ref | Expression |
---|---|
euop2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
euop2 | ⊢ (∃!𝑥∃𝑦(𝑥 = 〈𝐴, 𝑦〉 ∧ 𝜑) ↔ ∃!𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 4859 | . 2 ⊢ 〈𝐴, 𝑦〉 ∈ V | |
2 | euop2.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | 2 | moop2 4891 | . 2 ⊢ ∃*𝑦 𝑥 = 〈𝐴, 𝑦〉 |
4 | 1, 3 | euxfr2 3358 | 1 ⊢ (∃!𝑥∃𝑦(𝑥 = 〈𝐴, 𝑦〉 ∧ 𝜑) ↔ ∃!𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃!weu 2458 Vcvv 3173 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: dfac5lem1 8829 |
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