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Mirrors > Home > NFE 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 2489 | . 2 ⊢ ℲyA | |
2 | 1 | r2exf 2650 | 1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 ∈ wcel 1710 ∃wrex 2615 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 |
This theorem is referenced by: reean 2777 elxpk2 4197 evenfinex 4503 oddfinex 4504 rnoprab2 5577 rnmpt2 5717 lecex 6115 mucnc 6131 |
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