MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reueq1 Structured version   Visualization version   GIF version

Theorem reueq1 3323
Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
Assertion
Ref Expression
reueq1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reueq1
StepHypRef Expression
1 nfcv 2941 . 2 𝑥𝐴
2 nfcv 2941 . 2 𝑥𝐵
31, 2reueq1f 3319 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator