Theorem rabexd 4257

Description: Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4258. (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 4255	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
4	2, 3	syl 14	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ V)
5	1, 4	eqeltrid 2319	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  {crab 2524  Vcvv 2813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-sep 4228

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2815  df-in 3217  df-ss 3224

This theorem is referenced by:  rabex2  4258  2omapfi  7271  hashfibclem  11206  psrbasg  14829  psrelbas  14830  psr0cl  14836  psr0lid  14837  psrnegcl  14838  psrlinv  14839  psrgrp  14840  psr1clfi  14843  mplvalcoe  14845  incistruhgr  16085  clwwlkng  16400