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Mirrors > Home > MPE Home > Th. List > csbab | Structured version Visualization version GIF version |
Description: Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 19-Aug-2018.) |
Ref | Expression |
---|---|
csbab | ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2800 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) | |
2 | sbsbc 3775 | . . . 4 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) | |
3 | 1, 2 | bitri 277 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) |
4 | sbccom 3853 | . . . 4 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | df-clab 2800 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
6 | sbsbc 3775 | . . . . . 6 ⊢ ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) | |
7 | 5, 6 | bitri 277 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) |
8 | 7 | sbcbii 3828 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑) |
9 | 4, 8 | bitr4i 280 | . . 3 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝑧 ∈ {𝑦 ∣ 𝜑}) |
10 | sbcel2 4366 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ {𝑦 ∣ 𝜑} ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑}) | |
11 | 3, 9, 10 | 3bitrri 300 | . 2 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑}) |
12 | 11 | eqriv 2818 | 1 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 [wsb 2065 ∈ wcel 2110 {cab 2799 [wsbc 3771 ⦋csb 3882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-nul 4291 |
This theorem is referenced by: csbsng 4637 csbuni 4859 csbxp 5644 csbdm 5760 csbwrdg 13889 abfmpeld 30393 abfmpel 30394 csbwrecsg 34602 csboprabg 34605 csbfinxpg 34663 csbingVD 41211 csbsngVD 41220 csbxpgVD 41221 csbrngVD 41223 csbunigVD 41225 csbfv12gALTVD 41226 |
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