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| Mirrors > Home > MPE Home > Th. List > inab | Structured version Visualization version GIF version | ||
| Description: Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
| Ref | Expression |
|---|---|
| inab | ⊢ ({𝑥 ∣ 𝜑} ∩ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ 𝜓)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sban 2080 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) | |
| 2 | df-clab 2715 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ 𝜓)} ↔ [𝑦 / 𝑥](𝜑 ∧ 𝜓)) | |
| 3 | df-clab 2715 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
| 4 | df-clab 2715 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
| 5 | 3, 4 | anbi12i 628 | . . 3 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) |
| 6 | 1, 2, 5 | 3bitr4ri 304 | . 2 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 ∧ 𝜓)}) |
| 7 | 6 | ineqri 4212 | 1 ⊢ ({𝑥 ∣ 𝜑} ∩ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ 𝜓)} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 [wsb 2064 ∈ wcel 2108 {cab 2714 ∩ cin 3950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 |
| This theorem is referenced by: inrab 4316 inrab2 4317 dfrab3 4319 orduniss2 7853 ssenen 9191 hashf1lem2 14495 symgsubmefmnd 19416 ballotlem2 34491 fmla0disjsuc 35403 dfiota3 35924 bj-inrab 36928 ptrest 37626 sticksstones22 42169 diophin 42783 |
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