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Theorem rabss3d 28542
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 1829 . 2 𝑥𝜑
2 nfrab1 3098 . 2 𝑥{𝑥𝐴𝜓}
3 nfrab1 3098 . 2 𝑥{𝑥𝐵𝜓}
4 rabss3d.1 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)
5 simprr 791 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜓)
64, 5jca 552 . . . 4 ((𝜑 ∧ (𝑥𝐴𝜓)) → (𝑥𝐵𝜓))
76ex 448 . . 3 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜓)))
8 rabid 3094 . . 3 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
9 rabid 3094 . . 3 (𝑥 ∈ {𝑥𝐵𝜓} ↔ (𝑥𝐵𝜓))
107, 8, 93imtr4g 283 . 2 (𝜑 → (𝑥 ∈ {𝑥𝐴𝜓} → 𝑥 ∈ {𝑥𝐵𝜓}))
111, 2, 3, 10ssrd 3572 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  {crab 2899  wss 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-in 3546  df-ss 3553
This theorem is referenced by:  xpinpreima2  29087
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