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Theorem zfrep4 5173
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 2803 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
21anbi1i 626 . . . 4 ((𝑥 ∈ {𝑥𝜑} ∧ 𝜓) ↔ (𝜑𝜓))
32exbii 1849 . . 3 (∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓) ↔ ∃𝑥(𝜑𝜓))
43abbii 2886 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} = {𝑦 ∣ ∃𝑥(𝜑𝜓)}
5 nfab1 2976 . . . . 5 𝑥{𝑥𝜑}
6 zfrep4.1 . . . . 5 {𝑥𝜑} ∈ V
7 zfrep4.2 . . . . . 6 (𝜑 → ∃𝑧𝑦(𝜓𝑦 = 𝑧))
81, 7sylbi 220 . . . . 5 (𝑥 ∈ {𝑥𝜑} → ∃𝑧𝑦(𝜓𝑦 = 𝑧))
95, 6, 8zfrepclf 5171 . . . 4 𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓))
10 abeq2 2944 . . . . 5 (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} ↔ ∀𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)))
1110exbii 1849 . . . 4 (∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} ↔ ∃𝑧𝑦(𝑦𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)))
129, 11mpbir 234 . . 3 𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)}
1312issetri 3487 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥𝜑} ∧ 𝜓)} ∈ V
144, 13eqeltrri 2909 1 {𝑦 ∣ ∃𝑥(𝜑𝜓)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115  {cab 2799  Vcvv 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473
This theorem is referenced by:  zfpair  5295  cshwsexa  14165
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