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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssrabi | Structured version Visualization version GIF version |
Description: Inference of restricted abstraction subclass from implication. (Contributed by Peter Mazsa, 26-Oct-2022.) |
Ref | Expression |
---|---|
ssrabi.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
ssrabi | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrabi.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
3 | 2 | ss2rabi 3937 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2050 {crab 3086 ⊆ wss 3823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rab 3091 df-in 3830 df-ss 3837 |
This theorem is referenced by: refrelsredund4 35312 |
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