Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rabss3d Structured version   Visualization version   GIF version

Theorem rabss3d 28736
 Description: Subclass law for restricted abstraction. (Contributed by Thierry Arnoux, 25-Sep-2017.)
Hypothesis
Ref Expression
rabss3d.1 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)
Assertion
Ref Expression
rabss3d (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rabss3d
StepHypRef Expression
1 nfv 1830 . 2 𝑥𝜑
2 nfrab1 3099 . 2 𝑥{𝑥𝐴𝜓}
3 nfrab1 3099 . 2 𝑥{𝑥𝐵𝜓}
4 rabss3d.1 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)
5 simprr 792 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜓)
64, 5jca 553 . . . 4 ((𝜑 ∧ (𝑥𝐴𝜓)) → (𝑥𝐵𝜓))
76ex 449 . . 3 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜓)))
8 rabid 3095 . . 3 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
9 rabid 3095 . . 3 (𝑥 ∈ {𝑥𝐵𝜓} ↔ (𝑥𝐵𝜓))
107, 8, 93imtr4g 284 . 2 (𝜑 → (𝑥 ∈ {𝑥𝐴𝜓} → 𝑥 ∈ {𝑥𝐵𝜓}))
111, 2, 3, 10ssrd 3573 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  {crab 2900   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-in 3547  df-ss 3554 This theorem is referenced by:  xpinpreima2  29281
 Copyright terms: Public domain W3C validator