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Mirrors > Home > ILE Home > Th. List > rabeqf | GIF version |
Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) |
Ref | Expression |
---|---|
rabeqf.1 | ⊢ Ⅎ𝑥𝐴 |
rabeqf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
rabeqf | ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeqf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | rabeqf.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfeq 2327 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | eleq2 2241 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 4 | anbi1d 465 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
6 | 3, 5 | abbid 2294 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)}) |
7 | df-rab 2464 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
8 | df-rab 2464 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
9 | 6, 7, 8 | 3eqtr4g 2235 | 1 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {cab 2163 Ⅎwnfc 2306 {crab 2459 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 |
This theorem is referenced by: rabeqif 2728 rabeq 2729 |
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