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Theorem cbveu 2671
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker cbveuw 2669 when possible. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbveu.1 𝑦𝜑
cbveu.2 𝑥𝜓
cbveu.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbveu (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Proof of Theorem cbveu
StepHypRef Expression
1 cbveu.1 . . 3 𝑦𝜑
21sb8eu 2664 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3 cbveu.2 . . . 4 𝑥𝜓
4 cbveu.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2524 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
65eubii 2648 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
72, 6bitri 278 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wnf 1785  [wsb 2069  ∃!weu 2631
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-10 2143  ax-11 2159  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632
This theorem is referenced by:  cbvreu  3397  cbvreucsf  3875
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