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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 5356 | . 2 ⊢ 〈𝐴, 𝑦〉 ∈ V | |
2 | euop2.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | 2 | moop2 5392 | . 2 ⊢ ∃*𝑦 𝑥 = 〈𝐴, 𝑦〉 |
4 | 1, 3 | euxfr2w 3711 | 1 ⊢ (∃!𝑥∃𝑦(𝑥 = 〈𝐴, 𝑦〉 ∧ 𝜑) ↔ ∃!𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃!weu 2653 Vcvv 3494 〈cop 4573 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 |
This theorem is referenced by: dfac5lem1 9549 |
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