Theorem rabexgf 45025

Description: A version of rabexg 5295 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 3409	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
2		simpl 482	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴)
3	2	ss2abi 4033	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴}
4		rabexgf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
5	4	abid2f 2923	. . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
6	3, 5	sseqtri 3998	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
7	1, 6	eqsstri 3996	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
8		ssexg 5281	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
9	7, 8	mpan 690	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  {cab 2708  Ⅎwnfc 2877  {crab 3408  Vcvv 3450   ⊆ wss 3917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-in 3924  df-ss 3934

This theorem is referenced by:  rabexf  45135  stoweidlem27  46032  stoweidlem35  46040