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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 1529 | . 2 ⊢ ⊤ | |
2 | rabxfr.1 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
3 | rabxfr.2 | . . 3 ⊢ Ⅎ𝑦𝐶 | |
4 | rabxfr.3 | . . . 4 ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) | |
5 | 4 | adantl 482 | . . 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 5216 | . 2 ⊢ ((⊤ ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) |
9 | 1, 8 | mpan 686 | 1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1525 ⊤wtru 1526 ∈ wcel 2083 Ⅎwnfc 2935 {crab 3111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-rab 3116 df-v 3442 |
This theorem is referenced by: (None) |
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