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Mirrors > Home > NFE Home > Th. List > 2reu5lem2 | GIF version |
Description: Lemma for 2reu5 3045. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Ref | Expression |
---|---|
2reu5lem2 | ⊢ (∀x ∈ A ∃*y ∈ B φ ↔ ∀x∃*y(x ∈ A ∧ y ∈ B ∧ φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 2623 | . . 3 ⊢ (∃*y ∈ B φ ↔ ∃*y(y ∈ B ∧ φ)) | |
2 | 1 | ralbii 2639 | . 2 ⊢ (∀x ∈ A ∃*y ∈ B φ ↔ ∀x ∈ A ∃*y(y ∈ B ∧ φ)) |
3 | df-ral 2620 | . . 3 ⊢ (∀x ∈ A ∃*y(y ∈ B ∧ φ) ↔ ∀x(x ∈ A → ∃*y(y ∈ B ∧ φ))) | |
4 | moanimv 2262 | . . . . . 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 | mobii 2240 | . . . . 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 | albii 1566 | . . 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: → wi 4 ↔ wb 176 ∧ wa 358 ∧ w3a 934 ∀wal 1540 ∈ wcel 1710 ∃*wmo 2205 ∀wral 2615 ∃*wrmo 2618 |
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-ral 2620 df-rmo 2623 |
This theorem is referenced by: 2reu5lem3 3044 |
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