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Theorem ssintrab 4936
Description: Subclass of the intersection of a restricted class abstraction. (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 3407 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21inteqi 4915 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
32sseq2i 3977 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
4 impexp 452 . . . 4 (((𝑥𝐵𝜑) → 𝐴𝑥) ↔ (𝑥𝐵 → (𝜑𝐴𝑥)))
54albii 1822 . . 3 (∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥) ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
6 ssintab 4930 . . 3 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥))
7 df-ral 3062 . . 3 (∀𝑥𝐵 (𝜑𝐴𝑥) ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
85, 6, 73bitr4i 303 . 2 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
93, 8bitri 275 1 (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wcel 2107  {cab 2710  wral 3061  {crab 3406  wss 3914   cint 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3449  df-in 3921  df-ss 3931  df-int 4912
This theorem is referenced by:  knatar  7306  harval2  9941  pwfseqlem3  10604  fldgenid  32140  ldgenpisyslem3  32828  topjoin  34890
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