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Mirrors > Home > ILE Home > Th. List > rabbidva2 | GIF version |
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
Ref | Expression |
---|---|
rabbidva2.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) |
Ref | Expression |
---|---|
rabbidva2 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbidva2.1 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) | |
2 | 1 | abbidv 2288 | . 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}) |
3 | df-rab 2457 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
4 | df-rab 2457 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)} | |
5 | 2, 3, 4 | 3eqtr4g 2228 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 {cab 2156 {crab 2452 |
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 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-rab 2457 |
This theorem is referenced by: (None) |
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