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Theorem ab2rexex2 7855
Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7844. (Contributed by NM, 20-Sep-2011.)
Hypotheses
Ref Expression
ab2rexex2.1 𝐴 ∈ V
ab2rexex2.2 𝐵 ∈ V
ab2rexex2.3 {𝑧𝜑} ∈ V
Assertion
Ref Expression
ab2rexex2 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
Distinct variable groups:   𝑥,𝑧,𝐴   𝑦,𝑧,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑦)   𝐵(𝑥)

Proof of Theorem ab2rexex2
StepHypRef Expression
1 ab2rexex2.1 . 2 𝐴 ∈ V
2 ab2rexex2.2 . . 3 𝐵 ∈ V
3 ab2rexex2.3 . . 3 {𝑧𝜑} ∈ V
42, 3abrexex2 7844 . 2 {𝑧 ∣ ∃𝑦𝐵 𝜑} ∈ V
51, 4abrexex2 7844 1 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2104  {cab 2713  wrex 3071  Vcvv 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-v 3439  df-in 3899  df-ss 3909  df-uni 4845  df-iun 4933
This theorem is referenced by:  brdom7disj  10333  brdom6disj  10334  lineset  37794
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