Theorem rabexgf 44417

Description: A version of rabexg 5337 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypothesis

Ref	Expression
rabexgf.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
rabexgf	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Proof of Theorem rabexgf

Step	Hyp	Ref	Expression
1		df-rab 3431	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		simpl 481	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴)
3	2	ss2abi 4063	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴}
4		rabexgf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5	4	abid2f 2933	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
6	3, 5	sseqtri 4018	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
7	1, 6	eqsstri 4016	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
8		ssexg 5327	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
9	7, 8	mpan 688	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098  {cab 2705  Ⅎwnfc 2879  {crab 3430  Vcvv 3473   ⊆ wss 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3431  df-v 3475  df-in 3956  df-ss 3966

This theorem is referenced by:  rabexf  44531  stoweidlem27  45444  stoweidlem35  45452