Theorem abexex 7954

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 3065	. . . 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 1842	. . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	1, 4	bitr4i 278	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥𝜑)
6	5	abbii 2796	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
7		abexex.1	. . 3 ⊢ 𝐴 ∈ V
8		abexex.3	. . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V
9	7, 8	abrexex2 7952	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V
10	6, 9	eqeltrri 2824	1 ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1773   ∈ wcel 2098  {cab 2703  ∃wrex 3064  Vcvv 3468

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-in 3950  df-ss 3960  df-uni 4903  df-iun 4992

This theorem is referenced by:  brdom7disj  10525  brdom6disj  10526