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Mirrors > Home > ILE Home > Th. List > rabbiia | GIF version |
Description: Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.) |
Ref | Expression |
---|---|
rabbiia.1 | ⊢ (x ∈ A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rabbiia | ⊢ {x ∈ A ∣ φ} = {x ∈ A ∣ ψ} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbiia.1 | . . . 4 ⊢ (x ∈ A → (φ ↔ ψ)) | |
2 | 1 | pm5.32i 427 | . . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ)) |
3 | 2 | abbii 2150 | . 2 ⊢ {x ∣ (x ∈ A ∧ φ)} = {x ∣ (x ∈ A ∧ ψ)} |
4 | df-rab 2309 | . 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)} | |
5 | df-rab 2309 | . 2 ⊢ {x ∈ A ∣ ψ} = {x ∣ (x ∈ A ∧ ψ)} | |
6 | 3, 4, 5 | 3eqtr4i 2067 | 1 ⊢ {x ∈ A ∣ φ} = {x ∈ A ∣ ψ} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 {cab 2023 {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-rab 2309 |
This theorem is referenced by: bm2.5ii 4188 fndmdifcom 5216 cauappcvgprlemladdru 6628 cauappcvgprlemladdrl 6629 cauappcvgpr 6634 caucvgprlemcl 6647 caucvgprlemladdrl 6649 caucvgpr 6653 ioopos 8589 |
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