Theorem rabexd 5226

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 5227. (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 5224	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
5	1, 4	eqeltrid 2842	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  {crab 3065  Vcvv 3408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883

This theorem is referenced by:  rabex2  5227  zorn2lem1  10110  sylow2a  19008  evlslem6  21041  mhpaddcl  21091  mhpvscacl  21094  mretopd  21989  cusgrexilem1  27527  vtxdgf  27559  mntoval  30979  tocycval  31094  prmidlval  31326  evlsbagval  39985  mhpind  39993  stoweidlem35  43251  stoweidlem50  43266  stoweidlem57  43273  stoweidlem59  43275  subsaliuncllem  43571  subsaliuncl  43572  smflimlem1  43978  smflimlem2  43979  smflimlem3  43980  smflimlem6  43983  smfrec  43995  smfpimcclem  44012  smfsuplem1  44016  smfinflem  44022  smflimsuplem1  44025  smflimsuplem2  44026  smflimsuplem3  44027  smflimsuplem4  44028  smflimsuplem5  44029  smflimsuplem7  44031  fvmptrab  44456  prproropen  44633