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Mirrors > Home > NFE Home > Th. List > rexcomf | GIF version |
Description: Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
ralcomf.1 | ⊢ ℲyA |
ralcomf.2 | ⊢ ℲxB |
Ref | Expression |
---|---|
rexcomf | ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 437 | . . . . 5 ⊢ ((x ∈ A ∧ y ∈ B) ↔ (y ∈ B ∧ x ∈ A)) | |
2 | 1 | anbi1i 676 | . . . 4 ⊢ (((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ((y ∈ B ∧ x ∈ A) ∧ φ)) |
3 | 2 | 2exbii 1583 | . . 3 ⊢ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ∃x∃y((y ∈ B ∧ x ∈ A) ∧ φ)) |
4 | excom 1741 | . . 3 ⊢ (∃x∃y((y ∈ B ∧ x ∈ A) ∧ φ) ↔ ∃y∃x((y ∈ B ∧ x ∈ A) ∧ φ)) | |
5 | 3, 4 | bitri 240 | . 2 ⊢ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ∃y∃x((y ∈ B ∧ x ∈ A) ∧ φ)) |
6 | ralcomf.1 | . . 3 ⊢ ℲyA | |
7 | 6 | r2exf 2650 | . 2 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ)) |
8 | ralcomf.2 | . . 3 ⊢ ℲxB | |
9 | 8 | r2exf 2650 | . 2 ⊢ (∃y ∈ B ∃x ∈ A φ ↔ ∃y∃x((y ∈ B ∧ x ∈ A) ∧ φ)) |
10 | 5, 7, 9 | 3bitr4i 268 | 1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A 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: rexcom 2772 |
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