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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 2681 | . . 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 2678 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
6 | 3, 5 | eqtrd 2173 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 {crab 2421 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rab 2426 |
This theorem is referenced by: elfvmptrab1 5523 mpoxopoveq 6145 supeq123d 6886 phival 11925 dfphi2 11932 cldval 12307 neifval 12348 cnfval 12402 cnpfval 12403 cnprcl2k 12414 hmeofvalg 12511 ispsmet 12531 ismet 12552 isxmet 12553 blfvalps 12593 cncfval 12767 |
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