Theorem reueq1 3323

Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
reueq1	⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reueq1

Step	Hyp	Ref	Expression
1		nfcv 2941	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2941	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	reueq1f 3319	1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653  ∃!wreu 3091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-mo 2591  df-eu 2609  df-cleq 2792  df-clel 2795  df-nfc 2930  df-reu 3096

This theorem is referenced by:  reueqd  3331  lubfval  17293  glbfval  17306  uspgredg2vlem  26456  uspgredg2v  26457  isfrgr  27607  frgr1v  27620  nfrgr2v  27621  frgr3v  27624  1vwmgr  27625  3vfriswmgr  27627  isplig  27856  hdmap14lem4a  37892  hdmap14lem15  37903