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| Mirrors > Home > MPE Home > Th. List > spsbcd | Structured version Visualization version 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 2068 and rspsbc 3879. (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 3801 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜓 → [𝐴 / 𝑥]𝜓)) | |
| 4 | 1, 2, 3 | sylc 65 | 1 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 ∈ wcel 2108 [wsbc 3788 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 | 
| This theorem is referenced by: ovmpodxf 7583 ex-natded9.26 30438 spsbcdi 38125 ovmpordxf 48255 | 
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