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Theorem reueq1 3360
 Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2142, ax-11 2158, and ax-12 2175. (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 2878 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 632 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
32eubidv 2647 . 2 (𝐴 = 𝐵 → (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐵𝜑)))
4 df-reu 3113 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
5 df-reu 3113 . 2 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
63, 4, 53bitr4g 317 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃!weu 2628  ∃!wreu 3108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2598  df-eu 2629  df-cleq 2791  df-clel 2870  df-reu 3113 This theorem is referenced by:  reueqd  3366  lubfval  17583  glbfval  17596  uspgredg2vlem  27023  uspgredg2v  27024  isfrgr  28055  frgr1v  28066  nfrgr2v  28067  frgr3v  28070  1vwmgr  28071  3vfriswmgr  28073  isplig  28269  hdmap14lem4a  39186  hdmap14lem15  39197
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