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Theorem rabssf 39616
Description: Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
rabssf.1 𝑥𝐵
Assertion
Ref Expression
rabssf ({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))

Proof of Theorem rabssf
StepHypRef Expression
1 df-rab 2950 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
21sseq1i 3662 . 2 ({𝑥𝐴𝜑} ⊆ 𝐵 ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐵)
3 rabssf.1 . . 3 𝑥𝐵
43abssf 39609 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐵 ↔ ∀𝑥((𝑥𝐴𝜑) → 𝑥𝐵))
5 impexp 461 . . . 4 (((𝑥𝐴𝜑) → 𝑥𝐵) ↔ (𝑥𝐴 → (𝜑𝑥𝐵)))
65albii 1787 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝑥𝐵) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥𝐵)))
7 df-ral 2946 . . 3 (∀𝑥𝐴 (𝜑𝑥𝐵) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝑥𝐵)))
86, 7bitr4i 267 . 2 (∀𝑥((𝑥𝐴𝜑) → 𝑥𝐵) ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))
92, 4, 83bitri 286 1 ({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wcel 2030  {cab 2637  wnfc 2780  wral 2941  {crab 2945  wss 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-in 3614  df-ss 3621
This theorem is referenced by:  rabssd  39646  supminfxr2  40012  preimageiingt  41251  preimaleiinlt  41252  smfmullem4  41322
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