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Theorem ssrabf 44620
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 3419 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
21sseq2i 4006 . 2 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ 𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
3 ssrabf.1 . . 3 𝑥𝐵
43ssabf 44606 . 2 (𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
5 ssrabf.2 . . . . 5 𝑥𝐴
63, 5dfss3f 3968 . . . 4 (𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
76anbi1i 622 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐵 𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
8 r19.26 3100 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
9 df-ral 3051 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
107, 8, 93bitr2ri 299 . 2 (∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
112, 4, 103bitri 296 1 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wcel 2098  {cab 2702  wnfc 2875  wral 3050  {crab 3418  wss 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rab 3419  df-ss 3961
This theorem is referenced by:  ssrabdf  44621  supminfxr2  44989  pimgtmnf2  46240  smfmullem4  46320  smflimsuplem7  46352
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