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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 2353 and spsbc 3448. See also rspsbca 3519 and rspcsbela 4006. (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 | cbvralsv 3182 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑) | |
2 | dfsbcq2 3438 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | rspcv 3305 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) |
4 | 1, 3 | syl5bi 232 | 1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 1880 ∈ wcel 1990 ∀wral 2912 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 df-sbc 3436 |
This theorem is referenced by: rspsbca 3519 sbcth2 3523 rspcsbela 4006 riota5f 6636 riotass2 6638 fzrevral 12425 fprodcllemf 14688 rspsbc2 38744 truniALT 38751 rspsbc2VD 39090 truniALTVD 39114 trintALTVD 39116 trintALT 39117 |
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