Theorem abexex 8012

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 3077	. . . 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 1846	. . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	1, 4	bitr4i 278	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥𝜑)
6	5	abbii 2812	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
7		abexex.1	. . 3 ⊢ 𝐴 ∈ V
8		abexex.3	. . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V
9	7, 8	abrexex2 8010	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V
10	6, 9	eqeltrri 2841	1 ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∃wrex 3076  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-v 3490  df-ss 3993  df-uni 4932  df-iun 5017

This theorem is referenced by:  brdom7disj  10600  brdom6disj  10601