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Theorem ssintrab 4498
Description: Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
Assertion
Ref Expression
ssintrab (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem ssintrab
StepHypRef Expression
1 df-rab 2920 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21inteqi 4477 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
32sseq2i 3628 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
4 impexp 462 . . . 4 (((𝑥𝐵𝜑) → 𝐴𝑥) ↔ (𝑥𝐵 → (𝜑𝐴𝑥)))
54albii 1746 . . 3 (∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥) ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
6 ssintab 4492 . . 3 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥))
7 df-ral 2916 . . 3 (∀𝑥𝐵 (𝜑𝐴𝑥) ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
85, 6, 73bitr4i 292 . 2 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
93, 8bitri 264 1 (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1480  wcel 1989  {cab 2607  wral 2911  {crab 2915  wss 3572   cint 4473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rab 2920  df-v 3200  df-in 3579  df-ss 3586  df-int 4474
This theorem is referenced by:  knatar  6604  harval2  8820  pwfseqlem3  9479  ldgenpisyslem3  30213  topjoin  32344
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