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Mirrors > Home > MPE Home > Th. List > rspcsbela | Structured version Visualization version GIF version |
Description: Special case related to rspsbc 3518. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
Ref | Expression |
---|---|
rspcsbela | ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspsbc 3518 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → [𝐴 / 𝑥]𝐶 ∈ 𝐷)) | |
2 | sbcel1g 3987 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝐶 ∈ 𝐷 ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷)) | |
3 | 1, 2 | sylibd 229 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷)) |
4 | 3 | imp 445 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ∈ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ∀wral 2912 [wsbc 3435 ⦋csb 3533 |
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-fal 1489 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 df-csb 3534 df-dif 3577 df-nul 3916 |
This theorem is referenced by: el2mpt2csbcl 7250 mptnn0fsupp 12797 mptnn0fsuppr 12799 fsumzcl2 14469 fsummsnunz 14483 fsumsplitsnun 14484 fsummsnunzOLD 14485 fsumsplitsnunOLD 14486 modfsummodslem1 14524 fprodmodd 14728 sumeven 15110 sumodd 15111 gsummpt1n0 18364 gsummptnn0fz 18382 telgsumfzslem 18385 telgsumfzs 18386 telgsums 18390 mptscmfsupp0 18928 coe1fzgsumdlem 19671 gsummoncoe1 19674 evl1gsumdlem 19720 madugsum 20449 iunmbl2 23325 gsumvsca1 29782 gsumvsca2 29783 esum2dlem 30154 esumiun 30156 iblsplitf 40186 fsummsndifre 41342 fsumsplitsndif 41343 fsummmodsndifre 41344 fsummmodsnunz 41345 |
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