Theorem abexex 7984

Description: A condition where a class abstraction continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)

Hypotheses

Ref	Expression
abexex.1	⊢ 𝐴 ∈ V
abexex.2	⊢ (𝜑 → 𝑥 ∈ 𝐴)
abexex.3	⊢ {𝑦 ∣ 𝜑} ∈ V

Assertion

Ref	Expression
abexex	⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem abexex

Step	Hyp	Ref	Expression
1		df-rex 3087	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		abexex.2	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
3	2	pm4.71ri 570	. . . . 5 ⊢ (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
4	3	exbii 1880	. . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	1, 4	bitr4i 281	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥𝜑)
6	5	abbii 2827	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
7		abexex.1	. . 3 ⊢ 𝐴 ∈ V
8		abexex.3	. . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V
9	7, 8	abrexex2 7982	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V
10	6, 9	eqeltrri 2857	1 ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3896  df-uni 4867  df-iun 4952

This theorem is used by:  brdom7disj  10614  brdom6disj  10615