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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 1749 and rspsbc 2995. (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 2924 | . 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 1330 ∈ wcel 1481 [wsbc 2913 |
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 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 df-sbc 2914 |
This theorem is referenced by: ovmpodxf 5904 |
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