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Mirrors > Home > NFE Home > Th. List > r19.12sn | GIF version |
Description: Special case of r19.12 2727 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
r19.12sn.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
r19.12sn | ⊢ (∃x ∈ {A}∀y ∈ B φ ↔ ∀y ∈ B ∃x ∈ {A}φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.12sn.1 | . 2 ⊢ A ∈ V | |
2 | sbcralg 3120 | . . 3 ⊢ (A ∈ V → ([̣A / x]̣∀y ∈ B φ ↔ ∀y ∈ B [̣A / x]̣φ)) | |
3 | rexsns 3764 | . . 3 ⊢ (A ∈ V → (∃x ∈ {A}∀y ∈ B φ ↔ [̣A / x]̣∀y ∈ B φ)) | |
4 | rexsns 3764 | . . . 4 ⊢ (A ∈ V → (∃x ∈ {A}φ ↔ [̣A / x]̣φ)) | |
5 | 4 | ralbidv 2634 | . . 3 ⊢ (A ∈ V → (∀y ∈ B ∃x ∈ {A}φ ↔ ∀y ∈ B [̣A / x]̣φ)) |
6 | 2, 3, 5 | 3bitr4d 276 | . 2 ⊢ (A ∈ V → (∃x ∈ {A}∀y ∈ B φ ↔ ∀y ∈ B ∃x ∈ {A}φ)) |
7 | 1, 6 | ax-mp 5 | 1 ⊢ (∃x ∈ {A}∀y ∈ B φ ↔ ∀y ∈ B ∃x ∈ {A}φ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615 Vcvv 2859 [̣wsbc 3046 {csn 3737 |
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-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-sn 3741 |
This theorem is referenced by: (None) |
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