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Theorem zfrep4 4887
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 2712 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21anbi1i 733 . . . 4 ((𝑥 ∈ {𝑥𝜑} ∧ 𝜓) ↔ (𝜑𝜓))
32exbii 1887 . . 3 (∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓) ↔ ∃𝑥(𝜑𝜓))
43abbii 2841 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} = {𝑦 ∣ ∃𝑥(𝜑𝜓)}
5 nfab1 2868 . . . . 5 𝑥{𝑥𝜑}
6 zfrep4.1 . . . . 5 {𝑥𝜑} ∈ V
7 zfrep4.2 . . . . . 6 (𝜑 → ∃𝑧𝑦(𝜓𝑦 = 𝑧))
81, 7sylbi 207 . . . . 5 (𝑥 ∈ {𝑥𝜑} → ∃𝑧𝑦(𝜓𝑦 = 𝑧))
95, 6, 8zfrepclf 4885 . . . 4 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓))
10 abeq2 2834 . . . . 5 (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)))
1110exbii 1887 . . . 4 (∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)))
129, 11mpbir 221 . . 3 𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)}
1312issetri 3314 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} ∈ V
144, 13eqeltrri 2800 1 {𝑦 ∣ ∃𝑥(𝜑𝜓)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1594   = wceq 1596  wex 1817  wcel 2103  {cab 2710  Vcvv 3304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306
This theorem is referenced by:  zfpair  5009  cshwsexa  13691
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