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Mirrors > Home > ILE Home > Th. List > rspc2ev | GIF version |
Description: 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.) |
Ref | Expression |
---|---|
rspc2v.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
rspc2v.2 | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspc2ev | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝜓) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspc2v.2 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) | |
2 | 1 | rspcev 2789 | . . . 4 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝜓) → ∃𝑦 ∈ 𝐷 𝜒) |
3 | 2 | anim2i 339 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝐵 ∈ 𝐷 ∧ 𝜓)) → (𝐴 ∈ 𝐶 ∧ ∃𝑦 ∈ 𝐷 𝜒)) |
4 | 3 | 3impb 1177 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝜓) → (𝐴 ∈ 𝐶 ∧ ∃𝑦 ∈ 𝐷 𝜒)) |
5 | rspc2v.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
6 | 5 | rexbidv 2438 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ 𝐷 𝜑 ↔ ∃𝑦 ∈ 𝐷 𝜒)) |
7 | 6 | rspcev 2789 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ∃𝑦 ∈ 𝐷 𝜒) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 𝜑) |
8 | 4, 7 | syl 14 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝜓) → ∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 |
This theorem is referenced by: rspc3ev 2806 opelxp 4569 rspceov 5813 2dom 6699 apreim 8365 addcn2 11079 mulcn2 11081 divalglemnn 11615 bezoutlema 11687 bezoutlemb 11688 txuni2 12425 txopn 12434 txdis 12446 txdis1cn 12447 xmettxlem 12678 |
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