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Mirrors > Home > NFE Home > Th. List > sbceqal | GIF version |
Description: Set theory version of sbeqal1 in set.mm. (Contributed by Andrew Salmon, 28-Jun-2011.) |
Ref | Expression |
---|---|
sbceqal | ⊢ (A ∈ V → (∀x(x = A → x = B) → A = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbc 3058 | . 2 ⊢ (A ∈ V → (∀x(x = A → x = B) → [̣A / x]̣(x = A → x = B))) | |
2 | sbcimg 3087 | . . 3 ⊢ (A ∈ V → ([̣A / x]̣(x = A → x = B) ↔ ([̣A / x]̣x = A → [̣A / x]̣x = B))) | |
3 | eqid 2353 | . . . . 5 ⊢ A = A | |
4 | eqsbc3 3085 | . . . . 5 ⊢ (A ∈ V → ([̣A / x]̣x = A ↔ A = A)) | |
5 | 3, 4 | mpbiri 224 | . . . 4 ⊢ (A ∈ V → [̣A / x]̣x = A) |
6 | pm5.5 326 | . . . 4 ⊢ ([̣A / x]̣x = A → (([̣A / x]̣x = A → [̣A / x]̣x = B) ↔ [̣A / x]̣x = B)) | |
7 | 5, 6 | syl 15 | . . 3 ⊢ (A ∈ V → (([̣A / x]̣x = A → [̣A / x]̣x = B) ↔ [̣A / x]̣x = B)) |
8 | eqsbc3 3085 | . . 3 ⊢ (A ∈ V → ([̣A / x]̣x = B ↔ A = B)) | |
9 | 2, 7, 8 | 3bitrd 270 | . 2 ⊢ (A ∈ V → ([̣A / x]̣(x = A → x = B) ↔ A = B)) |
10 | 1, 9 | sylibd 205 | 1 ⊢ (A ∈ V → (∀x(x = A → x = B) → A = B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 = wceq 1642 ∈ wcel 1710 [̣wsbc 3046 |
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-v 2861 df-sbc 3047 |
This theorem is referenced by: sbeqalb 3098 |
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