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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rababw | Structured version Visualization version GIF version | ||
| Description: A weak version of rabab 3512 not using df-clel 2816 nor df-v 3482 (but requiring ax-ext 2708) nor ax-12 2177. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-rababw.1 | ⊢ 𝜓 |
| Ref | Expression |
|---|---|
| bj-rababw | ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-rab 3437 | . 2 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)} | |
| 2 | bj-rababw.1 | . . . . 5 ⊢ 𝜓 | |
| 3 | 2 | vexw 2720 | . . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓} |
| 4 | 3 | biantrur 530 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)) |
| 5 | 4 | abbii 2809 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)} |
| 6 | 1, 5 | eqtr4i 2768 | 1 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 {crab 3436 |
| 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-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-rab 3437 |
| This theorem is referenced by: (None) |
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