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Theorem cbveu 2607
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker cbveuw 2606, cbveuvw 2605 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 2600 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3 cbveu.2 . . . 4 𝑥𝜓
4 cbveu.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2507 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
65eubii 2585 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
72, 6bitri 275 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wnf 1783  [wsb 2064  ∃!weu 2568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569
This theorem is referenced by:  cbvreu  3428  cbvreucsf  3943
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