![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rabxfr | Structured version Visualization version GIF version |
Description: Membership in a restricted class abstraction after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the formula defining the class abstraction. (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 1538 | . 2 ⊢ ⊤ | |
2 | rabxfr.1 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
3 | rabxfr.2 | . . 3 ⊢ Ⅎ𝑦𝐶 | |
4 | rabxfr.3 | . . . 4 ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) | |
5 | 4 | adantl 480 | . . 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 5413 | . 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 205 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 Ⅎwnfc 2876 {crab 3419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-rab 3420 df-v 3464 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |