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Theorem reueq1 3405
Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2136, ax-11 2151, and ax-12 2167. (Revised by Steven Nguyen, 30-Apr-2023.)
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 eleq2 2898 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 629 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
32eubidv 2665 . 2 (𝐴 = 𝐵 → (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐵𝜑)))
4 df-reu 3142 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
5 df-reu 3142 . 2 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
63, 4, 53bitr4g 315 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  ∃!weu 2646  ∃!wreu 3137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-mo 2615  df-eu 2647  df-cleq 2811  df-clel 2890  df-reu 3142
This theorem is referenced by:  reueqd  3417  lubfval  17576  glbfval  17589  uspgredg2vlem  26932  uspgredg2v  26933  isfrgr  27966  frgr1v  27977  nfrgr2v  27978  frgr3v  27981  1vwmgr  27982  3vfriswmgr  27984  isplig  28180  hdmap14lem4a  38887  hdmap14lem15  38898
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