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Mirrors > Home > ILE Home > Th. List > spsbc | GIF version |
Description: Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1705 and rspsbc 2921. (Contributed by NM, 16-Jan-2004.) |
Ref | Expression |
---|---|
spsbc | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 1705 | . . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
2 | sbsbc 2844 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | sylib 120 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
4 | dfsbcq 2842 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
5 | 3, 4 | syl5ib 152 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
6 | 5 | vtocleg 2690 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1287 = wceq 1289 ∈ wcel 1438 [wsb 1692 [wsbc 2840 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-v 2621 df-sbc 2841 |
This theorem is referenced by: spsbcd 2852 sbcth 2853 sbcthdv 2854 sbceqal 2894 sbcimdv 2904 csbiebt 2967 csbexga 3967 |
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