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Mirrors > Home > NFE Home > Th. List > r2exf | GIF version |
Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
r2alf.1 | ⊢ ℲyA |
Ref | Expression |
---|---|
r2exf | ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2620 | . 2 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x(x ∈ A ∧ ∃y ∈ B φ)) | |
2 | r2alf.1 | . . . . . 6 ⊢ ℲyA | |
3 | 2 | nfcri 2483 | . . . . 5 ⊢ Ⅎy x ∈ A |
4 | 3 | 19.42 1880 | . . . 4 ⊢ (∃y(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ (x ∈ A ∧ ∃y(y ∈ B ∧ φ))) |
5 | anass 630 | . . . . 5 ⊢ (((x ∈ A ∧ y ∈ B) ∧ φ) ↔ (x ∈ A ∧ (y ∈ B ∧ φ))) | |
6 | 5 | exbii 1582 | . . . 4 ⊢ (∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ∃y(x ∈ A ∧ (y ∈ B ∧ φ))) |
7 | df-rex 2620 | . . . . 5 ⊢ (∃y ∈ B φ ↔ ∃y(y ∈ B ∧ φ)) | |
8 | 7 | anbi2i 675 | . . . 4 ⊢ ((x ∈ A ∧ ∃y ∈ B φ) ↔ (x ∈ A ∧ ∃y(y ∈ B ∧ φ))) |
9 | 4, 6, 8 | 3bitr4i 268 | . . 3 ⊢ (∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ (x ∈ A ∧ ∃y ∈ B φ)) |
10 | 9 | exbii 1582 | . 2 ⊢ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ∃x(x ∈ A ∧ ∃y ∈ B φ)) |
11 | 1, 10 | bitr4i 243 | 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 Ⅎwnfc 2476 ∃wrex 2615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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: r2ex 2652 rexcomf 2770 |
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