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Theorem eubidOLD 2643
 Description: Obsolete proof of eubid 2625 as of 14-Oct-2022. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eubidOLD.1 𝑥𝜑
eubidOLD.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
eubidOLD (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Proof of Theorem eubidOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eubidOLD.1 . . . 4 𝑥𝜑
2 eubidOLD.2 . . . . 5 (𝜑 → (𝜓𝜒))
32bibi1d 335 . . . 4 (𝜑 → ((𝜓𝑥 = 𝑦) ↔ (𝜒𝑥 = 𝑦)))
41, 3albid 2257 . . 3 (𝜑 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜒𝑥 = 𝑦)))
54exbidv 2017 . 2 (𝜑 → (∃𝑦𝑥(𝜓𝑥 = 𝑦) ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦)))
6 eu6 2611 . 2 (∃!𝑥𝜓 ↔ ∃𝑦𝑥(𝜓𝑥 = 𝑦))
7 eu6 2611 . 2 (∃!𝑥𝜒 ↔ ∃𝑦𝑥(𝜒𝑥 = 𝑦))
85, 6, 73bitr4g 306 1 (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651  ∃wex 1875  Ⅎwnf 1879  ∃!weu 2606 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880  df-mo 2590  df-eu 2607 This theorem is referenced by: (None)
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