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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 2802 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) | |
2 | sbsbc 3778 | . . . 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 3856 | . . . 4 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | df-clab 2802 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
6 | sbsbc 3778 | . . . . . 6 ⊢ ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) | |
7 | 5, 6 | bitri 277 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) |
8 | 7 | sbcbii 3831 | . . . 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 4369 | . . 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 2820 | 1 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 [wsb 2069 ∈ wcel 2114 {cab 2801 [wsbc 3774 ⦋csb 3885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-nul 4294 |
This theorem is referenced by: csbsng 4646 csbuni 4869 csbxp 5652 csbdm 5768 csbwrdg 13897 abfmpeld 30401 abfmpel 30402 csbwrecsg 34610 csboprabg 34613 csbfinxpg 34671 csbingVD 41225 csbsngVD 41234 csbxpgVD 41235 csbrngVD 41237 csbunigVD 41239 csbfv12gALTVD 41240 |
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