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Mirrors > Home > ILE Home > Th. List > rabbidva | GIF version |
Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.) |
Ref | Expression |
---|---|
rabbidva.1 | ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ)) |
Ref | Expression |
---|---|
rabbidva | ⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ A ∣ χ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbidva.1 | . . 3 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ)) | |
2 | 1 | ralrimiva 2386 | . 2 ⊢ (φ → ∀x ∈ A (ψ ↔ χ)) |
3 | rabbi 2481 | . 2 ⊢ (∀x ∈ A (ψ ↔ χ) ↔ {x ∈ A ∣ ψ} = {x ∈ A ∣ χ}) | |
4 | 2, 3 | sylib 127 | 1 ⊢ (φ → {x ∈ A ∣ ψ} = {x ∈ A ∣ χ}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∀wral 2300 {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-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 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-ral 2305 df-rab 2309 |
This theorem is referenced by: rabbidv 2543 rabeqbidva 2547 rabbi2dva 3139 rabxfrd 4167 onsucmin 4198 seinxp 4354 fniniseg2 5232 fnniniseg2 5233 f1oresrab 5272 |
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