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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 2741. (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 2737 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∈ wcel 2160 {crab 2472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-rab 2477 |
This theorem is referenced by: dfdif3 3260 suplocexpr 7742 dfrhm2 13465 dmtopon 13920 |
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