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Mirrors > Home > ILE Home > Th. List > rabbii | GIF version |
Description: Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 2678. (Contributed by Peter Mazsa, 1-Nov-2019.) |
Ref | Expression |
---|---|
rabbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
rabbii | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbii.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | a1i 9 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) |
3 | 2 | rabbiia 2674 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1332 ∈ wcel 1481 {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-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 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-rab 2426 |
This theorem is referenced by: dfdif3 3191 suplocexpr 7557 dmtopon 12229 |
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