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Mirrors > Home > MPE Home > Th. List > cbvrab | Structured version Visualization version GIF version |
Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.) |
Ref | Expression |
---|---|
cbvrab.1 | ⊢ Ⅎ𝑥𝐴 |
cbvrab.2 | ⊢ Ⅎ𝑦𝐴 |
cbvrab.3 | ⊢ Ⅎ𝑦𝜑 |
cbvrab.4 | ⊢ Ⅎ𝑥𝜓 |
cbvrab.5 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrab | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑) | |
2 | cbvrab.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2758 | . . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
4 | nfs1v 2437 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
5 | 3, 4 | nfan 1828 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) |
6 | eleq1 2689 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
7 | sbequ12 2111 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
8 | 6, 7 | anbi12d 747 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))) |
9 | 1, 5, 8 | cbvab 2746 | . . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)} |
10 | cbvrab.2 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
11 | 10 | nfcri 2758 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 |
12 | cbvrab.3 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
13 | 12 | nfsb 2440 | . . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
14 | 11, 13 | nfan 1828 | . . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) |
15 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓) | |
16 | eleq1 2689 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
17 | sbequ 2376 | . . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
18 | cbvrab.4 | . . . . . . 7 ⊢ Ⅎ𝑥𝜓 | |
19 | cbvrab.5 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
20 | 18, 19 | sbie 2408 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
21 | 17, 20 | syl6bb 276 | . . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓)) |
22 | 16, 21 | anbi12d 747 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓))) |
23 | 14, 15, 22 | cbvab 2746 | . . 3 ⊢ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} |
24 | 9, 23 | eqtri 2644 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} |
25 | df-rab 2921 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
26 | df-rab 2921 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} | |
27 | 24, 25, 26 | 3eqtr4i 2654 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 [wsb 1880 ∈ wcel 1990 {cab 2608 Ⅎwnfc 2751 {crab 2916 |
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-rab 2921 |
This theorem is referenced by: cbvrabv 3199 elrabsf 3474 tfis 7054 cantnflem1 8586 scottexs 8750 scott0s 8751 elmptrab 21630 bnj1534 30923 scottexf 33976 scott0f 33977 eq0rabdioph 37340 rexrabdioph 37358 rexfrabdioph 37359 elnn0rabdioph 37367 dvdsrabdioph 37374 binomcxplemdvsum 38554 fnlimcnv 39899 fnlimabslt 39911 stoweidlem34 40251 stoweidlem59 40276 pimltmnf2 40911 pimgtpnf2 40917 pimltpnf2 40923 issmff 40943 smfpimltxrmpt 40967 smfpreimagtf 40976 smflim 40985 smfpimgtxr 40988 smfpimgtxrmpt 40992 smflim2 41012 smflimsup 41034 smfliminf 41037 |
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