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Mirrors > Home > ILE Home > Th. List > rabeqbidv | GIF version |
Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.) |
Ref | Expression |
---|---|
rabeqbidv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
rabeqbidv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rabeqbidv | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqbidv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | rabeq 2673 | . . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) |
4 | rabeqbidv.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
5 | 4 | rabbidv 2670 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
6 | 3, 5 | eqtrd 2170 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 {crab 2418 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rab 2423 |
This theorem is referenced by: elfvmptrab1 5508 mpoxopoveq 6130 supeq123d 6871 phival 11878 dfphi2 11885 cldval 12257 neifval 12298 cnfval 12352 cnpfval 12353 cnprcl2k 12364 hmeofvalg 12461 ispsmet 12481 ismet 12502 isxmet 12503 blfvalps 12543 cncfval 12717 |
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