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Mirrors > Home > MPE Home > Th. List > eqrrabd | Structured version Visualization version GIF version |
Description: Deduce equality with a restricted abstraction. (Contributed by Thierry Arnoux, 11-Apr-2024.) |
Ref | Expression |
---|---|
eqrrabd.1 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
eqrrabd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓)) |
Ref | Expression |
---|---|
eqrrabd | ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1913 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfcv 2908 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfrab1 3464 | . 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓} | |
4 | eqrrabd.1 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
5 | 4 | sseld 4007 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) |
6 | 5 | pm4.71rd 562 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) |
7 | eqrrabd.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓)) | |
8 | 7 | pm5.32da 578 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) |
9 | 6, 8 | bitrd 279 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) |
10 | rabid 3465 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) | |
11 | 9, 10 | bitr4di 289 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})) |
12 | 1, 2, 3, 11 | eqrd 4028 | 1 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {crab 3443 ⊆ wss 3976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-rab 3444 df-ss 3993 |
This theorem is referenced by: usgrexmpl2nb0 47846 usgrexmpl2nb1 47847 usgrexmpl2nb2 47848 usgrexmpl2nb3 47849 usgrexmpl2nb4 47850 usgrexmpl2nb5 47851 |
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