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Mirrors > Home > MPE Home > Th. List > rabeqi | Structured version Visualization version GIF version |
Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 3482. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2141 and ax-11 2156. (Revised by Gino Giotto, 20-Aug-2023.) |
Ref | Expression |
---|---|
rabeqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
rabeqi | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | 1 | nfth 1798 | . . . 4 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
3 | eleq2 2901 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
4 | 3 | anbi1d 631 | . . . 4 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
5 | 2, 4 | abbid 2887 | . . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
6 | df-rab 3147 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
7 | df-rab 3147 | . . 3 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
8 | 5, 6, 7 | 3eqtr4g 2881 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
9 | 1, 8 | ax-mp 5 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 {crab 3142 |
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-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-rab 3147 |
This theorem is referenced by: rabrabi 3494 f1ossf1o 6885 hsmex2 9849 iooval2 12765 fzval2 12889 phimullem 16110 pmtrsn 18641 dsmmbas2 20875 qtopres 22300 uvtxval 27163 cusgredg 27200 cffldtocusgr 27223 vtxdginducedm1 27319 finsumvtxdg2size 27326 konigsbergiedgw 28021 extwwlkfab 28125 satf0 32614 prjspeclsp 39255 k0004val0 40497 smflimlem4 43043 smfliminf 43098 |
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