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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 3444. (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 3433 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
2 | rabeqcda.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
3 | 2 | ex 413 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) |
4 | 3 | pm4.71d 562 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) |
5 | 4 | eqabdv 2867 | . 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}) |
6 | 1, 5 | eqtr4id 2791 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2709 {crab 3432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 |
This theorem is referenced by: rabeqc 3444 cmnbascntr 19675 lrold 27399 prjcrv0 41457 mreclat 47700 |
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