Theorem rabexd 5334

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5335. (Contributed by AV, 16-Jul-2019.)

Hypotheses

Ref	Expression
rabexd.1	⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}
rabexd.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
rabexd	⊢ (𝜑 → 𝐵 ∈ V)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem rabexd

Step	Hyp	Ref	Expression
1		rabexd.1	. 2 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}
2		rabexd.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		rabexg 5332	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
5	1, 4	eqeltrid 2838	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475

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-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966

This theorem is referenced by:  rabex2  5335  zorn2lem1  10491  sylow2a  19487  evlslem6  21644  mhpaddcl  21694  mhpvscacl  21697  mretopd  22596  cusgrexilem1  28727  vtxdgf  28759  mntoval  32183  tocycval  32298  prmidlval  32586  evlsvvval  41183  evlsbagval  41186  mhpind  41214  stoweidlem35  44799  stoweidlem50  44814  stoweidlem57  44821  stoweidlem59  44823  subsaliuncllem  45121  subsaliuncl  45122  smflimlem1  45535  smflimlem2  45536  smflimlem3  45537  smflimlem6  45540  smfrec  45553  smfpimcclem  45571  smfsuplem1  45575  smfinflem  45581  smflimsuplem1  45584  smflimsuplem2  45585  smflimsuplem3  45586  smflimsuplem4  45587  smflimsuplem5  45588  smflimsuplem7  45590  fvmptrab  46048  prproropen  46224