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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-inrab2 | Structured version Visualization version GIF version |
Description: Shorter proof of inrab 4099. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-inrab2 | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-inrab 33417 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ (𝐴 ∩ 𝐴) ∣ (𝜑 ∧ 𝜓)} | |
2 | nfv 2010 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
3 | inidm 4018 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → (𝐴 ∩ 𝐴) = 𝐴) |
5 | 2, 4 | bj-rabeqd 33409 | . . 3 ⊢ (⊤ → {𝑥 ∈ (𝐴 ∩ 𝐴) ∣ (𝜑 ∧ 𝜓)} = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)}) |
6 | 5 | mptru 1661 | . 2 ⊢ {𝑥 ∈ (𝐴 ∩ 𝐴) ∣ (𝜑 ∧ 𝜓)} = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
7 | 1, 6 | eqtri 2821 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 = wceq 1653 ⊤wtru 1654 {crab 3093 ∩ cin 3768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-in 3776 |
This theorem is referenced by: (None) |
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