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Mirrors > Home > MPE Home > Th. List > spsbc | 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. This is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51. See also stdpc4 2064 and rspsbc 3859. (Contributed by NM, 16-Jan-2004.) |
Ref | Expression |
---|---|
spsbc | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc4 2064 | . . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
2 | sbsbc 3773 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | sylib 219 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
4 | dfsbcq 3771 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
5 | 3, 4 | syl5ib 245 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
6 | 5 | vtocleg 3578 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 = wceq 1528 [wsb 2060 ∈ wcel 2105 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-sbc 3770 |
This theorem is referenced by: spsbcd 3783 sbcth 3784 sbcthdv 3785 sbceqal 3832 sbcimdv 3840 csbiebt 3909 csbexg 5205 pm14.18 40637 sbcbi 40750 onfrALTlem3 40755 sbc3orgVD 41062 sbcbiVD 41087 csbingVD 41095 onfrALTlem3VD 41098 csbeq2gVD 41103 csbunigVD 41109 |
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