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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 1995 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑) | |
2 | cbvrab.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2907 | . . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
4 | nfs1v 2274 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
5 | 3, 4 | nfan 1980 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) |
6 | eleq1w 2833 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
7 | sbequ12 2267 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
8 | 6, 7 | anbi12d 616 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))) |
9 | 1, 5, 8 | cbvab 2895 | . . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)} |
10 | cbvrab.2 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
11 | 10 | nfcri 2907 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 |
12 | cbvrab.3 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
13 | 12 | nfsb 2590 | . . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
14 | 11, 13 | nfan 1980 | . . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) |
15 | nfv 1995 | . . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓) | |
16 | eleq1w 2833 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
17 | sbequ 2523 | . . . . . 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 2555 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
21 | 17, 20 | syl6bb 276 | . . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓)) |
22 | 16, 21 | anbi12d 616 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓))) |
23 | 14, 15, 22 | cbvab 2895 | . . 3 ⊢ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} |
24 | 9, 23 | eqtri 2793 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} |
25 | df-rab 3070 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
26 | df-rab 3070 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} | |
27 | 24, 25, 26 | 3eqtr4i 2803 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 Ⅎwnf 1856 [wsb 2049 ∈ wcel 2145 {cab 2757 Ⅎwnfc 2900 {crab 3065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-rab 3070 |
This theorem is referenced by: cbvrabv 3349 elrabsf 3626 f1ossf1o 6538 tfis 7201 cantnflem1 8750 scottexs 8914 scott0s 8915 elmptrab 21851 dlwwlknondlwlknonf1oOLD 27556 bnj1534 31261 scottexf 34308 scott0f 34309 eq0rabdioph 37866 rexrabdioph 37884 rexfrabdioph 37885 elnn0rabdioph 37893 dvdsrabdioph 37900 binomcxplemdvsum 39080 fnlimcnv 40417 fnlimabslt 40429 stoweidlem34 40768 stoweidlem59 40793 pimltmnf2 41431 pimgtpnf2 41437 pimltpnf2 41443 issmff 41463 smfpimltxrmpt 41487 smfpreimagtf 41496 smflim 41505 smfpimgtxr 41508 smfpimgtxrmpt 41512 smflim2 41532 smflimsup 41554 smfliminf 41557 |
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