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Theorem ssrabf 42365
Description: Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
ssrabf.1 𝑥𝐵
ssrabf.2 𝑥𝐴
Assertion
Ref Expression
ssrabf (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))

Proof of Theorem ssrabf
StepHypRef Expression
1 df-rab 3071 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
21sseq2i 3944 . 2 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ 𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
3 ssrabf.1 . . 3 𝑥𝐵
43ssabf 42351 . 2 (𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
5 ssrabf.2 . . . . 5 𝑥𝐴
63, 5dfss3f 3905 . . . 4 (𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
76anbi1i 627 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐵 𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
8 r19.26 3093 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
9 df-ral 3067 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
107, 8, 93bitr2ri 303 . 2 (∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
112, 4, 103bitri 300 1 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541  wcel 2111  {cab 2715  wnfc 2885  wral 3062  {crab 3066  wss 3880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rab 3071  df-v 3422  df-in 3887  df-ss 3897
This theorem is referenced by:  supminfxr2  42712  pimgtmnf2  43951  smfmullem4  44028  smflimsuplem7  44059
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