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

Theorem reueq1 3411
Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2130, ax-11 2147, and ax-12 2167. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2101. (Revised by Wolf Lammen, 12-Mar-2025.)
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 rexeq 3317 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜑))
2 rmoeq1 3410 . . 3 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
31, 2anbi12d 631 . 2 (𝐴 = 𝐵 → ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐴 𝜑) ↔ (∃𝑥𝐵 𝜑 ∧ ∃*𝑥𝐵 𝜑)))
4 reu5 3374 . 2 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐴 𝜑))
5 reu5 3374 . 2 (∃!𝑥𝐵 𝜑 ↔ (∃𝑥𝐵 𝜑 ∧ ∃*𝑥𝐵 𝜑))
63, 4, 53bitr4g 314 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wrex 3066  ∃!wreu 3370  ∃*wrmo 3371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-mo 2530  df-eu 2559  df-cleq 2720  df-rex 3067  df-rmo 3372  df-reu 3373
This theorem is referenced by:  reueqd  3415  lubfval  18335  glbfval  18348  uspgredg2vlem  29029  uspgredg2v  29030  isfrgr  30063  frgr1v  30074  nfrgr2v  30075  frgr3v  30078  1vwmgr  30079  3vfriswmgr  30081  isplig  30279  hdmap14lem4a  41338  hdmap14lem15  41349
  Copyright terms: Public domain W3C validator