Theorem reueq 1326

Description: Equality theorem for restricted uniqueness quantifier.

Assertion

Ref	Expression
reueq	⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ))

Distinct variable group(s):   x,A   x,B

Proof of Theorem reueq

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (y ∈ A → ∀x y ∈ A)
2		ax-17 925	. 2 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	reueqf 1323	1 ⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  ∃!wreu 1203

This theorem is referenced by:  reueqd 1329

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-9 799  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-eu 1009  df-cleq 1097  df-clel 1099  df-reu 1207

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