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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 2715 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) | |
2 | sbsbc 3687 | . . . 4 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) | |
3 | 1, 2 | bitri 278 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) |
4 | sbccom 3770 | . . . 4 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | |
5 | df-clab 2715 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
6 | sbsbc 3687 | . . . . . 6 ⊢ ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) | |
7 | 5, 6 | bitri 278 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) |
8 | 7 | sbcbii 3743 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑) |
9 | 4, 8 | bitr4i 281 | . . 3 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝑧 ∈ {𝑦 ∣ 𝜑}) |
10 | sbcel2 4316 | . . 3 ⊢ ([𝐴 / 𝑥]𝑧 ∈ {𝑦 ∣ 𝜑} ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑}) | |
11 | 3, 9, 10 | 3bitrri 301 | . 2 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑}) |
12 | 11 | eqriv 2733 | 1 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 [wsb 2072 ∈ wcel 2112 {cab 2714 [wsbc 3683 ⦋csb 3798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-nul 4224 |
This theorem is referenced by: csbsng 4610 csbuni 4836 csbxp 5632 csbdm 5751 csbwrdg 14064 abfmpeld 30665 abfmpel 30666 csbwrecsg 35184 csboprabg 35187 csbfinxpg 35245 csbingVD 42118 csbsngVD 42127 csbxpgVD 42128 csbrngVD 42130 csbunigVD 42132 csbfv12gALTVD 42133 |
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