Theorem euop2 1912

Description: Transfer existential uniqueness to second member of an ordered pair.

Assertion

Ref	Expression
euop2	⊢ (∃!x∃y(x = ⟨A, y⟩ ∧ φ) ↔ ∃!yφ)

Distinct variable group(s):   φ,x   x,A   x,y

Proof of Theorem euop2

Step	Hyp	Ref	Expression
1		opex 1893	. 2 ⊢ ⟨A, y⟩ ∈ V
2		moop2 1910	. 2 ⊢ ∃*y x = ⟨A, y⟩
3	1, 2	euxfr2 1435	1 ⊢ (∃!x∃y(x = ⟨A, y⟩ ∧ φ) ↔ ∃!yφ)

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678  ∃!weu 1007   = wceq 1091  ⟨cop 1810

This theorem is referenced by:  aceq5lem1 3558

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-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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