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| Mirrors > Home > MPE Home > Th. List > rabeq | Structured version Visualization version GIF version | ||
| Description: Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) Avoid ax-10 2141, ax-11 2157, ax-12 2177. (Revised by GG, 20-Aug-2023.) |
| Ref | Expression |
|---|---|
| rabeq | ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq2 2830 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
| 2 | 1 | anbi1d 631 | . 2 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
| 3 | 2 | rabbidva2 3438 | 1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 |
| This theorem is referenced by: rabeqdv 3452 difeq1 4119 ineq1 4213 ifeq1 4529 ifeq2 4530 elfvmptrab 7045 supp0 8190 supeq2 9488 oieq2 9553 scott0 9926 mrcfval 17651 ipoval 18575 mndpsuppss 18778 psgnfval 19518 rgspnval 20612 dsmmelbas 21759 psrval 21935 ltbval 22061 opsrval 22064 m1detdiag 22603 isptfin 23524 islocfin 23525 kqval 23734 incistruhgr 29096 uvtx0 29411 vtxdg0e 29492 1hevtxdg1 29524 hashecclwwlkn1 30096 umgrhashecclwwlk 30097 ordtrestNEW 33920 ordtrest2NEWlem 33921 omsval 34295 orrvcval4 34467 orrvcoel 34468 orrvccel 34469 funray 36141 fvray 36142 itg2addnclem2 37679 cntotbnd 37803 lcfr 41587 hlhilocv 41963 pellfundval 42891 elmnc 43148 rfovd 44014 fsovd 44021 fsovcnvlem 44026 ntrneibex 44086 dvnprodlem2 45962 dvnprodlem3 45963 dvnprod 45964 fvmptrab 47304 rmsuppss 48286 scmsuppss 48287 dmatALTbas 48318 |
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