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Mirrors > Home > ILE Home > Th. List > rabeq | GIF version |
Description: Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) |
Ref | Expression |
---|---|
rabeq | ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2258 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2258 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | rabeqf 2650 | 1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 {crab 2397 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rab 2402 |
This theorem is referenced by: rabeqdv 2654 rabeqbidv 2655 rabeqbidva 2656 difeq1 3157 ifeq1 3447 ifeq2 3448 elfvmptrab 5484 pmvalg 6521 unfiexmid 6774 ssfirab 6790 supeq2 6844 iooval2 9666 fzval2 9761 lcmval 11671 lcmcllem 11675 lcmledvds 11678 clsfval 12197 |
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