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Mirrors > Home > MPE Home > Th. List > abanssr | Structured version Visualization version GIF version |
Description: A class abstraction with a conjunction is a subset of the class abstraction with the right conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.) |
Ref | Expression |
---|---|
abanssr | ⊢ {𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜓) | |
2 | 1 | ss2abi 4000 | 1 ⊢ {𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 {cab 2715 ⊆ wss 3887 |
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 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-in 3894 df-ss 3904 |
This theorem is referenced by: hashf1lem1 14168 |
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