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Theorem cbveuwOLD 2627
 Description: Obsolete version of cbveuw 2626 as of 23-May-2024. (Contributed by NM, 25-Nov-1994.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbveuwOLD.1 𝑦𝜑
cbveuwOLD.2 𝑥𝜓
cbveuwOLD.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbveuwOLD (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbveuwOLD
StepHypRef Expression
1 cbveuwOLD.1 . . 3 𝑦𝜑
21sb8euv 2620 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3 cbveuwOLD.2 . . . 4 𝑥𝜓
4 cbveuwOLD.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbiev 2323 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
65eubii 2605 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
72, 6bitri 278 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1786  [wsb 2070  ∃!weu 2588 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 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589 This theorem is referenced by: (None)
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