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Theorem cbveuALT 2672
Description: Alternative proof of cbveu 2671. Since df-eu 2632 combines two other quantifiers, one can base this theorem on their associated 'change bounded variable' kind of theorems as well. (Contributed by Wolf Lammen, 5-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbveu.1 𝑦𝜑
cbveu.2 𝑥𝜓
cbveu.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbveuALT (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Proof of Theorem cbveuALT
StepHypRef Expression
1 cbveu.1 . . . 4 𝑦𝜑
2 cbveu.2 . . . 4 𝑥𝜓
3 cbveu.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvex 2409 . . 3 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
51, 2, 3cbvmo 2668 . . 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  wnf 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-10 2143  ax-11 2159  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632
This theorem is referenced by: (None)
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