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Theorem zfrep4 4973
Description: A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)
Hypotheses
Ref Expression
zfrep4.1 {𝑥𝜑} ∈ V
zfrep4.2 (𝜑 → ∃𝑧𝑦(𝜓𝑦 = 𝑧))
Assertion
Ref Expression
zfrep4 {𝑦 ∣ ∃𝑥(𝜑𝜓)} ∈ V
Distinct variable groups:   𝜑,𝑦,𝑧   𝜓,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem zfrep4
StepHypRef Expression
1 abid 2794 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21anbi1i 612 . . . 4 ((𝑥 ∈ {𝑥𝜑} ∧ 𝜓) ↔ (𝜑𝜓))
32exbii 1933 . . 3 (∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓) ↔ ∃𝑥(𝜑𝜓))
43abbii 2923 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} = {𝑦 ∣ ∃𝑥(𝜑𝜓)}
5 nfab1 2950 . . . . 5 𝑥{𝑥𝜑}
6 zfrep4.1 . . . . 5 {𝑥𝜑} ∈ V
7 zfrep4.2 . . . . . 6 (𝜑 → ∃𝑧𝑦(𝜓𝑦 = 𝑧))
81, 7sylbi 208 . . . . 5 (𝑥 ∈ {𝑥𝜑} → ∃𝑧𝑦(𝜓𝑦 = 𝑧))
95, 6, 8zfrepclf 4971 . . . 4 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓))
10 abeq2 2916 . . . . 5 (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)))
1110exbii 1933 . . . 4 (∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)))
129, 11mpbir 222 . . 3 𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)}
1312issetri 3404 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} ∈ V
144, 13eqeltrri 2882 1 {𝑦 ∣ ∃𝑥(𝜑𝜓)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2156  {cab 2792  Vcvv 3391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393
This theorem is referenced by:  zfpair  5094  cshwsexa  13794
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