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

Theorem reueq1 3170
 Description: Equality theorem for restricted uniqueness 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 2793 . 2 𝑥𝐴
2 nfcv 2793 . 2 𝑥𝐵
31, 2reueq1f 3166 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523  ∃!wreu 2943 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-eu 2502  df-cleq 2644  df-clel 2647  df-nfc 2782  df-reu 2948 This theorem is referenced by:  reueqd  3178  lubfval  17025  glbfval  17038  uspgredg2vlem  26160  uspgredg2v  26161  isfrgr  27238  frgr1v  27251  nfrgr2v  27252  frgr3v  27255  1vwmgr  27256  3vfriswmgr  27258  isplig  27458  hdmap14lem4a  37480  hdmap14lem15  37491
 Copyright terms: Public domain W3C validator