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Mirrors > Home > MPE Home > Th. List > rspsbc | Structured version Visualization version GIF version |
Description: Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 2072 and spsbc 3753. See also rspsbca 3837 and rspcsbela 4396. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
rspsbc | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvralsvw 3299 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑) | |
2 | dfsbcq2 3743 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | rspcv 3576 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) |
4 | 1, 3 | biimtrid 241 | 1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2068 ∈ wcel 2107 ∀wral 3061 [wsbc 3740 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-sbc 3741 |
This theorem is referenced by: rspsbca 3837 sbcth2 3841 rspcsbela 4396 riota5f 7343 riotass2 7345 fzrevral 13532 fprodcllemf 15846 rspsbc2 42904 truniALT 42911 rspsbc2VD 43225 truniALTVD 43248 trintALTVD 43250 trintALT 43251 |
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