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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 5555 | . . . . . . 7 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
2 | 1 | biimpi 217 | . . . . . 6 ⊢ (Rel 𝐴 → 𝐴 ⊆ (V × V)) |
3 | 2 | sseld 3963 | . . . . 5 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V))) |
4 | 3 | adantrd 492 | . . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ (V × V))) |
5 | 4 | pm4.71rd 563 | . . 3 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
6 | 5 | rexbidv2 3292 | . 2 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑))) |
7 | eleq1 2897 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝑥 ∈ 𝐴 ↔ 〈𝑦, 𝑧〉 ∈ 𝐴)) | |
8 | exopxfr2.1 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
9 | 7, 8 | anbi12d 630 | . . 3 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓))) |
10 | 9 | exopxfr 5707 | . 2 ⊢ (∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑧(〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓)) |
11 | 6, 10 | syl6bb 288 | 1 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∃wrex 3136 Vcvv 3492 ⊆ wss 3933 〈cop 4563 × cxp 5546 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-iun 4912 df-opab 5120 df-xp 5554 df-rel 5555 |
This theorem is referenced by: dvhopellsm 38133 |
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