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Mirrors > Home > NFE Home > Th. List > 2reu5lem1 | GIF version |
Description: Lemma for 2reu5 3044. Note that ∃!x ∈ A∃!y ∈ Bφ does not mean "there is exactly one x in A and exactly one y in B such that φ holds;" see comment for 2eu5 2288. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Ref | Expression |
---|---|
2reu5lem1 | ⊢ (∃!x ∈ A ∃!y ∈ B φ ↔ ∃!x∃!y(x ∈ A ∧ y ∈ B ∧ φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 2621 | . . 3 ⊢ (∃!y ∈ B φ ↔ ∃!y(y ∈ B ∧ φ)) | |
2 | 1 | reubii 2797 | . 2 ⊢ (∃!x ∈ A ∃!y ∈ B φ ↔ ∃!x ∈ A ∃!y(y ∈ B ∧ φ)) |
3 | df-reu 2621 | . . 3 ⊢ (∃!x ∈ A ∃!y(y ∈ B ∧ φ) ↔ ∃!x(x ∈ A ∧ ∃!y(y ∈ B ∧ φ))) | |
4 | euanv 2265 | . . . . . 6 ⊢ (∃!y(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ (x ∈ A ∧ ∃!y(y ∈ B ∧ φ))) | |
5 | 4 | bicomi 193 | . . . . 5 ⊢ ((x ∈ A ∧ ∃!y(y ∈ B ∧ φ)) ↔ ∃!y(x ∈ A ∧ (y ∈ B ∧ φ))) |
6 | 3anass 938 | . . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ B ∧ φ) ↔ (x ∈ A ∧ (y ∈ B ∧ φ))) | |
7 | 6 | bicomi 193 | . . . . . 6 ⊢ ((x ∈ A ∧ (y ∈ B ∧ φ)) ↔ (x ∈ A ∧ y ∈ B ∧ φ)) |
8 | 7 | eubii 2213 | . . . . 5 ⊢ (∃!y(x ∈ A ∧ (y ∈ B ∧ φ)) ↔ ∃!y(x ∈ A ∧ y ∈ B ∧ φ)) |
9 | 5, 8 | bitri 240 | . . . 4 ⊢ ((x ∈ A ∧ ∃!y(y ∈ B ∧ φ)) ↔ ∃!y(x ∈ A ∧ y ∈ B ∧ φ)) |
10 | 9 | eubii 2213 | . . 3 ⊢ (∃!x(x ∈ A ∧ ∃!y(y ∈ B ∧ φ)) ↔ ∃!x∃!y(x ∈ A ∧ y ∈ B ∧ φ)) |
11 | 3, 10 | bitri 240 | . 2 ⊢ (∃!x ∈ A ∃!y(y ∈ B ∧ φ) ↔ ∃!x∃!y(x ∈ A ∧ y ∈ B ∧ φ)) |
12 | 2, 11 | bitri 240 | 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 ∧ w3a 934 ∈ wcel 1710 ∃!weu 2204 ∃!wreu 2616 |
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 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-reu 2621 |
This theorem is referenced by: 2reu5lem3 3043 |
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