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Mirrors > Home > ILE Home > Th. List > r2ex | GIF version |
Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.) |
Ref | Expression |
---|---|
r2ex | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2240 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | 1 | r2exf 2412 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1436 ∈ wcel 1448 ∃wrex 2376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rex 2381 |
This theorem is referenced by: reean 2557 rexiunxp 4619 rnoprab2 5787 genprndl 7230 genprndu 7231 genpdisj 7232 prmuloc 7275 mullocpr 7280 axcnre 7566 |
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