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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 4932 | . 2 ⊢ 〈𝐴, 𝑦〉 ∈ V | |
2 | euop2.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | 2 | moop2 4966 | . 2 ⊢ ∃*𝑦 𝑥 = 〈𝐴, 𝑦〉 |
4 | 1, 3 | euxfr2 3391 | 1 ⊢ (∃!𝑥∃𝑦(𝑥 = 〈𝐴, 𝑦〉 ∧ 𝜑) ↔ ∃!𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 Vcvv 3200 〈cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: dfac5lem1 8946 |
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