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Theorem rabss3d 30203
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 1906 . 2 𝑥𝜑
2 nfrab1 3382 . 2 𝑥{𝑥𝐴𝜓}
3 nfrab1 3382 . 2 𝑥{𝑥𝐵𝜓}
4 rabss3d.1 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝑥𝐵)
5 simprr 769 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜓)
64, 5jca 512 . . . 4 ((𝜑 ∧ (𝑥𝐴𝜓)) → (𝑥𝐵𝜓))
76ex 413 . . 3 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜓)))
8 rabid 3376 . . 3 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
9 rabid 3376 . . 3 (𝑥 ∈ {𝑥𝐵𝜓} ↔ (𝑥𝐵𝜓))
107, 8, 93imtr4g 297 . 2 (𝜑 → (𝑥 ∈ {𝑥𝐴𝜓} → 𝑥 ∈ {𝑥𝐵𝜓}))
111, 2, 3, 10ssrd 3969 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  {crab 3139  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-in 3940  df-ss 3949
This theorem is referenced by:  xpinpreima2  31049  reprss  31787  reprinfz1  31792
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