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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 2306 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2306 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | rabeqf 2712 | 1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 {crab 2446 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rab 2451 |
This theorem is referenced by: rabeqdv 2716 rabeqbidv 2717 rabeqbidva 2718 difeq1 3229 ifeq1 3519 ifeq2 3520 elfvmptrab 5576 pmvalg 6617 unfiexmid 6875 ssfirab 6891 supeq2 6946 iooval2 9843 fzval2 9939 lcmval 11981 lcmcllem 11985 lcmledvds 11988 clsfval 12659 |
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