Theorem abexex 7903

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 3057	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		abexex.2	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
3	2	pm4.71ri 560	. . . . 5 ⊢ (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
4	3	exbii 1849	. . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	1, 4	bitr4i 278	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥𝜑)
6	5	abbii 2798	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
7		abexex.1	. . 3 ⊢ 𝐴 ∈ V
8		abexex.3	. . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V
9	7, 8	abrexex2 7901	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V
10	6, 9	eqeltrri 2828	1 ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∃wrex 3056  Vcvv 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-v 3438  df-ss 3919  df-uni 4860  df-iun 4943

This theorem is referenced by:  brdom7disj  10419  brdom6disj  10420