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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 5045 | . . . . . . 7 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
2 | 1 | biimpi 205 | . . . . . 6 ⊢ (Rel 𝐴 → 𝐴 ⊆ (V × V)) |
3 | 2 | sseld 3567 | . . . . 5 ⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (V × V))) |
4 | 3 | adantrd 483 | . . . 4 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ (V × V))) |
5 | 4 | pm4.71rd 665 | . . 3 ⊢ (Rel 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ (V × V) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))) |
6 | 5 | rexbidv2 3030 | . 2 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑))) |
7 | eleq1 2676 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝑥 ∈ 𝐴 ↔ 〈𝑦, 𝑧〉 ∈ 𝐴)) | |
8 | exopxfr2.1 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
9 | 7, 8 | anbi12d 743 | . . 3 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓))) |
10 | 9 | exopxfr 5187 | . 2 ⊢ (∃𝑥 ∈ (V × V)(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑧(〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓)) |
11 | 6, 10 | syl6bb 275 | 1 ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(〈𝑦, 𝑧〉 ∈ 𝐴 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 〈cop 4131 × cxp 5036 Rel wrel 5043 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 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 df-iun 4457 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: dvhopellsm 35424 |
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