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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 3466 not using df-clel 2863 nor df-v 3439 (but requiring ax-ext 2769) nor ax-12 2141. (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 3114 | . 2 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)} | |
2 | bj-rababw.1 | . . . . 5 ⊢ 𝜓 | |
3 | 2 | vexw 2781 | . . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓} |
4 | 3 | biantrur 531 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)) |
5 | 4 | abbii 2861 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)} |
6 | 1, 5 | eqtr4i 2822 | 1 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1522 ∈ wcel 2081 {cab 2775 {crab 3109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-9 2091 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 df-sb 2043 df-clab 2776 df-cleq 2788 df-rab 3114 |
This theorem is referenced by: (None) |
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