![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2175 | . 2 ⊢ ℲxA | |
2 | nfcv 2175 | . 2 ⊢ ℲxB | |
3 | 1, 2 | rabeqf 2544 | 1 ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 {crab 2304 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 |
This theorem is referenced by: rabeqbidv 2546 rabeqbidva 2547 difeq1 3049 ifeq1 3328 ifeq2 3329 iooval2 8554 fzval2 8647 |
Copyright terms: Public domain | W3C validator |