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Theorem cbveu 2504
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
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 2502 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3 cbveu.2 . . . 4 𝑥𝜓
4 cbveu.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2407 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
65eubii 2491 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
72, 6bitri 264 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1705  [wsb 1877  ∃!weu 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473
This theorem is referenced by:  cbvmo  2505  cbvreu  3160  cbvreucsf  3552  tz6.12f  6174  f1ompt  6343  climeu  14228  initoeu2  16598
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