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Theorem rabss3d 30294
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 1916 . 2 𝑥𝜑
2 nfrab1 3375 . 2 𝑥{𝑥𝐴𝜓}
3 nfrab1 3375 . 2 𝑥{𝑥𝐵𝜓}
4 rabss3d.1 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)
5 simprr 772 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜓)
64, 5jca 515 . . . 4 ((𝜑 ∧ (𝑥𝐴𝜓)) → (𝑥𝐵𝜓))
76ex 416 . . 3 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜓)))
8 rabid 3369 . . 3 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
9 rabid 3369 . . 3 (𝑥 ∈ {𝑥𝐵𝜓} ↔ (𝑥𝐵𝜓))
107, 8, 93imtr4g 299 . 2 (𝜑 → (𝑥 ∈ {𝑥𝐴𝜓} → 𝑥 ∈ {𝑥𝐵𝜓}))
111, 2, 3, 10ssrd 3958 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  {crab 3137  wss 3919
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936
This theorem is referenced by:  xpinpreima2  31235  reprss  31973  reprinfz1  31978
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