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Mirrors > Home > MPE Home > Th. List > rabeqcda | Structured version Visualization version GIF version |
Description: When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3445. (Contributed by Steven Nguyen, 7-Jun-2023.) |
Ref | Expression |
---|---|
rabeqcda.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
Ref | Expression |
---|---|
rabeqcda | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3434 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
2 | rabeqcda.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
3 | 2 | ex 414 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) |
4 | 3 | pm4.71d 563 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) |
5 | 4 | eqabdv 2868 | . 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) |
6 | 1, 5 | eqtr4id 2792 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 {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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 |
This theorem is referenced by: rabeqc 3445 cmnbascntr 19673 lrold 27391 prjcrv0 41375 mreclat 47622 |
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