Theorem euabex 1869

Description: The abstraction of a wff with existential uniqueness exists.

Assertion

Ref	Expression
euabex	⊢ (∃!xφ → {x∣φ} ∈ V)

Proof of Theorem euabex

Step	Hyp	Ref	Expression
1		eumo 1037	. 2 ⊢ (∃!xφ → ∃*xφ)
2		moabex 1868	. 2 ⊢ (∃*xφ → {x∣φ} ∈ V)
3	1, 2	syl 12	1 ⊢ (∃!xφ → {x∣φ} ∈ V)

Syntax hints:   → wi 2  ∃!weu 1007  ∃*wmo 1008  {cab 1090   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  euuni 1954

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811

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