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| Mirrors > Home > MPE Home > Th. List > exopxfr2 | Structured version Visualization version GIF version | ||
| Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) |
| Ref | Expression |
|---|---|
| exopxfr2.1 | ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| exopxfr2 | ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-rel 5692 | . . . . . . 7 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
| 2 | 1 | biimpi 216 | . . . . . 6 ⊢ (Rel 𝐴 → 𝐴 ⊆ (V × V)) |
| 3 | 2 | sseld 3982 | . . . . 5 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V))) |
| 4 | 3 | adantrd 491 | . . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ (V × V))) |
| 5 | 4 | pm4.71rd 562 | . . 3 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
| 6 | 5 | rexbidv2 3175 | . 2 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑))) |
| 7 | eleq1 2829 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝑥 ∈ 𝐴 ↔ 〈𝑦, 𝑧〉 ∈ 𝐴)) | |
| 8 | exopxfr2.1 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
| 9 | 7, 8 | anbi12d 632 | . . 3 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓))) |
| 10 | 9 | exopxfr 5854 | . 2 ⊢ (∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑧(〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓)) |
| 11 | 6, 10 | bitrdi 287 | 1 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 〈cop 4632 × cxp 5683 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: dvhopellsm 41119 |
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