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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 2484 and rspsbc 3743. (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 3676 | . 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 1656 ∈ wcel 2166 [wsbc 3663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1662 df-ex 1881 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-v 3417 df-sbc 3664 |
This theorem is referenced by: ovmpt2dxf 7047 ex-natded9.26 27835 spsbcdi 34464 ovmpt2rdxf 42965 |
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