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Theorem cbveuw 2669
 Description: Version of cbveu 2671 with a disjoint variable condition, which does not require ax-10 2143, ax-13 2382. (Contributed by NM, 25-Nov-1994.) (Revised by Gino Giotto, 23-May-2024.)
Hypotheses
Ref Expression
cbveuw.1 𝑦𝜑
cbveuw.2 𝑥𝜓
cbveuw.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbveuw (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbveuw
StepHypRef Expression
1 cbveuw.1 . . . 4 𝑦𝜑
2 cbveuw.2 . . . 4 𝑥𝜓
3 cbveuw.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvexv1 2354 . . 3 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
51, 2, 3cbvmow 2666 . . 3 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
64, 5anbi12i 629 . 2 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
7 df-eu 2632 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
8 df-eu 2632 . 2 (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
96, 7, 83bitr4i 306 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1781  Ⅎwnf 1785  ∃*wmo 2599  ∃!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-11 2159  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-mo 2601  df-eu 2632 This theorem is referenced by:  cbvreuw  3392  tz6.12f  6673  f1ompt  6856  climeu  14907  initoeu2  17271
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