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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 2072 and rspsbc 3840. (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 3757 | . 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 1540 ∈ wcel 2107 [wsbc 3744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-sbc 3745 |
This theorem is referenced by: ovmpodxf 7510 ex-natded9.26 29405 spsbcdi 36606 ovmpordxf 46488 |
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