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Mirrors > Home > ILE Home > Th. List > rabeqi | GIF version |
Description: Equality theorem for restricted class abstractions. Inference form of rabeq 2722. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
rabeqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
rabeqi | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2312 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2312 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | rabeqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
4 | 1, 2, 3 | rabeqif 2721 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 {crab 2452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rab 2457 |
This theorem is referenced by: phimullem 12166 odzcllem 12183 odzdvds 12186 |
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