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Theorem ab2rexex2 7145
Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7133. (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 7133 . 2 {𝑧 ∣ ∃𝑦𝐵 𝜑} ∈ V
51, 4abrexex2 7133 1 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 1988  {cab 2606  wrex 2910  Vcvv 3195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884
This theorem is referenced by:  brdom7disj  9338  brdom6disj  9339  lineset  34843
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