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Mirrors > Home > ILE Home > Th. List > rabbi | GIF version |
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2669. (Contributed by NM, 25-Nov-2013.) |
Ref | Expression |
---|---|
rabbi | ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbi 2251 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)}) | |
2 | df-ral 2419 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒))) | |
3 | pm5.32 448 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) | |
4 | 3 | albii 1446 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
5 | 2, 4 | bitri 183 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
6 | df-rab 2423 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
7 | df-rab 2423 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)} | |
8 | 6, 7 | eqeq12i 2151 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)}) |
9 | 1, 5, 8 | 3bitr4i 211 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1329 = wceq 1331 ∈ wcel 1480 {cab 2123 ∀wral 2414 {crab 2418 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-ral 2419 df-rab 2423 |
This theorem is referenced by: rabbidva 2669 exmidonfinlem 7042 |
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