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Theorem ssintrab 4976
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 3434 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21inteqi 4955 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
32sseq2i 4025 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
4 impexp 450 . . . 4 (((𝑥𝐵𝜑) → 𝐴𝑥) ↔ (𝑥𝐵 → (𝜑𝐴𝑥)))
54albii 1816 . . 3 (∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥) ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
6 ssintab 4970 . . 3 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥))
7 df-ral 3060 . . 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 206  wa 395  wal 1535  wcel 2106  {cab 2712  wral 3059  {crab 3433  wss 3963   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-ss 3980  df-int 4952
This theorem is referenced by:  knatar  7377  harval2  10035  pwfseqlem3  10698  elrgspnlem4  33235  ldgenpisyslem3  34146  topjoin  36348
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