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Theorem reubidva 3323
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023.)
Hypothesis
Ref Expression
reubidva.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
reubidva (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem reubidva
StepHypRef Expression
1 reubidva.1 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
21pm5.32da 579 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
32eubidv 2587 . 2 (𝜑 → (∃!𝑥(𝑥𝐴𝜓) ↔ ∃!𝑥(𝑥𝐴𝜒)))
4 df-reu 3073 . 2 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥(𝑥𝐴𝜓))
5 df-reu 3073 . 2 (∃!𝑥𝐴 𝜒 ↔ ∃!𝑥(𝑥𝐴𝜒))
63, 4, 53bitr4g 314 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2107  ∃!weu 2569  ∃!wreu 3067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2541  df-eu 2570  df-reu 3073
This theorem is referenced by:  reubidv  3324  reuxfrd  3684  reuxfr1d  3686  exfo  6990  f1ofveu  7279  zmax  12694  zbtwnre  12695  rebtwnz  12696  icoshftf1o  13215  divalgb  16122  1arith2  16638  ply1divalg2  25312  addsq2reu  26597  addsqn2reu  26598  addsqrexnreu  26599  2sqreultlem  26604  2sqreunnltlem  26607  frgr2wwlkeu  28700  numclwwlk2lem1  28749  numclwlk2lem2f1o  28752  pjhtheu2  29787  reuxfrdf  30848  xrsclat  31298  xrmulc1cn  31889  poimirlem25  35811  hdmap14lem14  39902  prproropreud  44972  quad1  45083  requad1  45085  requad2  45086  itscnhlinecirc02p  46142
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