Theorem zfrep4 5247

Description: A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)

Hypotheses

Ref	Expression
zfrep4.1	⊢ {𝑥 ∣ 𝜑} ∈ V
zfrep4.2	⊢ (𝜑 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))

Assertion

Ref	Expression
zfrep4	⊢ {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} ∈ V

Distinct variable groups:   𝜑,𝑦,𝑧   𝜓,𝑧   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem zfrep4

Step	Hyp	Ref	Expression
1		abid 2747	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2	1	anbi1i 635	. . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓) ↔ (𝜑 ∧ 𝜓))
3	2	exbii 1871	. . 3 ⊢ (∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓))
4	3	abbii 2832	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} = {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)}
5		nfab1 2929	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
6		zfrep4.1	. . . . 5 ⊢ {𝑥 ∣ 𝜑} ∈ V
7		zfrep4.2	. . . . . 6 ⊢ (𝜑 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
8	1, 7	sylbi 220	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
9	5, 6, 8	zfrepclf 5245	. . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓))
10		eqabb 2904	. . . . 5 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)))
11	10	exbii 1871	. . . 4 ⊢ (∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)))
12	9, 11	mpbir 234	. . 3 ⊢ ∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)}
13	12	issetri 3476	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ∈ V
14	4, 13	eqeltrri 2862	1 ⊢ {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} ∈ V

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-v 3459

This theorem is referenced by:  zfpair  5382