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Mirrors > Home > MPE Home > Th. List > rabxfr | Structured version Visualization version 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 1537 | . 2 ⊢ ⊤ | |
2 | rabxfr.1 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
3 | rabxfr.2 | . . 3 ⊢ Ⅎ𝑦𝐶 | |
4 | rabxfr.3 | . . . 4 ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) | |
5 | 4 | adantl 484 | . . 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 5309 | . 2 ⊢ ((⊤ ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) |
9 | 1, 8 | mpan 688 | 1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 Ⅎwnfc 2961 {crab 3142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 |
This theorem is referenced by: (None) |
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