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Mirrors > Home > ILE Home > Th. List > rabxfr | GIF version |
Description: Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜑. (Contributed by NM, 10-Jun-2005.) |
Ref | Expression |
---|---|
rabxfr.1 | ⊢ Ⅎ𝑦𝐵 |
rabxfr.2 | ⊢ Ⅎ𝑦𝐶 |
rabxfr.3 | ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) |
rabxfr.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
rabxfr.5 | ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) |
Ref | Expression |
---|---|
rabxfr | ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1335 | . 2 ⊢ ⊤ | |
2 | rabxfr.1 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
3 | rabxfr.2 | . . 3 ⊢ Ⅎ𝑦𝐶 | |
4 | rabxfr.3 | . . . 4 ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) | |
5 | 4 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷) |
6 | rabxfr.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | rabxfr.5 | . . 3 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) | |
8 | 2, 3, 5, 6, 7 | rabxfrd 4385 | . 2 ⊢ ((⊤ ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) |
9 | 1, 8 | mpan 420 | 1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 Ⅎwnfc 2266 {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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 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-clel 2133 df-nfc 2268 df-ral 2419 df-rab 2423 df-v 2683 |
This theorem is referenced by: (None) |
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