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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 3497 not using df-clel 2804 nor df-v 3470 (but requiring ax-ext 2697) nor ax-12 2163. (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 3427 | . 2 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)} | |
2 | bj-rababw.1 | . . . . 5 ⊢ 𝜓 | |
3 | 2 | vexw 2709 | . . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓} |
4 | 3 | biantrur 530 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)) |
5 | 4 | abbii 2796 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)} |
6 | 1, 5 | eqtr4i 2757 | 1 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 {cab 2703 {crab 3426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-rab 3427 |
This theorem is referenced by: (None) |
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