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Mirrors > Home > ILE Home > Th. List > spsbcd | 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 1763 and rspsbc 3033. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Ref | Expression |
---|---|
spsbcd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
spsbcd.2 | ⊢ (𝜑 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
spsbcd | ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbcd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | spsbcd.2 | . 2 ⊢ (𝜑 → ∀𝑥𝜓) | |
3 | spsbc 2962 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓)) | |
4 | 1, 2, 3 | sylc 62 | 1 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1341 ∈ wcel 2136 [wsbc 2951 |
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 1435 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-v 2728 df-sbc 2952 |
This theorem is referenced by: ovmpodxf 5967 |
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