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Mirrors > Home > NFE Home > Th. List > ss2rabdv | GIF version |
Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.) |
Ref | Expression |
---|---|
ss2rabdv.1 | ⊢ ((φ ∧ x ∈ A) → (ψ → χ)) |
Ref | Expression |
---|---|
ss2rabdv | ⊢ (φ → {x ∈ A ∣ ψ} ⊆ {x ∈ A ∣ χ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2rabdv.1 | . . 3 ⊢ ((φ ∧ x ∈ A) → (ψ → χ)) | |
2 | 1 | ralrimiva 2697 | . 2 ⊢ (φ → ∀x ∈ A (ψ → χ)) |
3 | ss2rab 3342 | . 2 ⊢ ({x ∈ A ∣ ψ} ⊆ {x ∈ A ∣ χ} ↔ ∀x ∈ A (ψ → χ)) | |
4 | 2, 3 | sylibr 203 | 1 ⊢ (φ → {x ∈ A ∣ ψ} ⊆ {x ∈ A ∣ χ}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 ∀wral 2614 {crab 2618 ⊆ wss 3257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 |
This theorem is referenced by: (None) |
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