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Theorem cbveu 2643
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 2641 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3 cbveu.2 . . . 4 𝑥𝜓
4 cbveu.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2545 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
65eubii 2629 . 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 1857  [wsb 2046  ∃!weu 2607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611
This theorem is referenced by:  cbvmo  2644  cbvreu  3308  cbvreucsf  3708  tz6.12f  6373  f1ompt  6545  climeu  14485  initoeu2  16867
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