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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabidd | Structured version Visualization version GIF version |
Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
Ref | Expression |
---|---|
rabidd.1 | ⊢ (𝜑 → 𝑥 ∈ 𝐴) |
rabidd.2 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
rabidd | ⊢ (𝜑 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabidd.1 | . 2 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
2 | rabidd.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | rabid 3453 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) | |
4 | 1, 2, 3 | sylanbrc 584 | 1 ⊢ (𝜑 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 {crab 3433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 |
This theorem is referenced by: fsupdm 45493 finfdm 45497 |
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