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| 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 2067 and rspsbc 3878. (Contributed by NM, 16-Jan-2004.) | 
| Ref | Expression | 
|---|---|
| spsbc | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | stdpc4 2067 | . . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
| 2 | sbsbc 3791 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
| 3 | 1, 2 | sylib 218 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | 
| 4 | dfsbcq 3789 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
| 5 | 3, 4 | imbitrid 244 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | 
| 6 | 5 | vtocleg 3552 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 = wceq 1539 [wsb 2063 ∈ wcel 2107 [wsbc 3787 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-sbc 3788 | 
| This theorem is referenced by: spsbcd 3801 sbcth 3802 sbcthdv 3803 sbceqalOLD 3851 csbiebt 3927 csbexg 5309 pm14.18 44452 sbcbi 44564 onfrALTlem3 44569 sbc3orgVD 44876 sbcbiVD 44901 csbingVD 44909 onfrALTlem3VD 44912 csbeq2gVD 44917 csbunigVD 44923 | 
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