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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 1748 and rspsbc 2991. (Contributed by NM, 16-Jan-2004.) |
Ref | Expression |
---|---|
spsbc | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 1748 | . . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
2 | sbsbc 2913 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | sylib 121 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
4 | dfsbcq 2911 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
5 | 3, 4 | syl5ib 153 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
6 | 5 | vtocleg 2757 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 = wceq 1331 ∈ wcel 1480 [wsb 1735 [wsbc 2909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-v 2688 df-sbc 2910 |
This theorem is referenced by: spsbcd 2921 sbcth 2922 sbcthdv 2923 sbceqal 2964 sbcimdv 2974 csbiebt 3039 csbexga 4056 |
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