Theorem rabexd 5308

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5309. (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 5306	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
4	2, 3	syl 18	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
5	1, 4	eqeltrid 2868	1 ⊢ (𝜑 → 𝐵 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562

This theorem is used by:  rabex2  5309  zorn2lem1  10613  sylow2a  19884  prmidlval  21669  psrascl  22341  evlslem6  22445  evlsvvval  22457  mhmcompl  22485  mhmcoaddmpl  22487  mhpaddcl  22527  mretopd  23465  plngval  29312  angmgmval  29451  cusgrexilem1  30077  vtxdgf  30109  mntoval  33642  tocycval  33768  fxpval  33825  selvply1rhmlemb  34262  extvfvcl  34279  isprimroot  43409  primrootsunit1  43413  unitscyglem1  43511  evlsbagval  43880  mhpind  43888  stoweidlem35  47318  stoweidlem50  47333  stoweidlem57  47340  stoweidlem59  47342  subsaliuncllem  47640  subsaliuncl  47641  smflimlem1  48054  smflimlem2  48055  smflimlem3  48056  smflimlem6  48059  smfrec  48072  smfpimcclem  48090  smfsuplem1  48094  smfinflem  48100  smflimsuplem1  48103  smflimsuplem2  48104  smflimsuplem3  48105  smflimsuplem4  48106  smflimsuplem5  48107  smflimsuplem7  48109  fvmptrab  48635  prproropen  48863  stgrvtx  49325  stgriedg  49326  gpgvtx  49414  gpgiedg  49415