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Theorem nfeu1ALT 2674
 Description: Alternate proof of nfeu1 2673. This illustrates the systematic way of proving nonfreeness in a defined expression: consider the definiens as a tree whose nodes are its subformulas, and prove by tree-induction nonfreeness of each node, starting from the leaves (generally using nfv 1915 or nf* theorems for previously defined expressions) and up to the root. Here, the definiens is a conjunction of two previously defined expressions, which automatically yields the present proof. (Contributed by BJ, 2-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nfeu1ALT 𝑥∃!𝑥𝜑

Proof of Theorem nfeu1ALT
StepHypRef Expression
1 df-eu 2653 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2 nfe1 2154 . . 3 𝑥𝑥𝜑
3 nfmo1 2640 . . 3 𝑥∃*𝑥𝜑
42, 3nfan 1900 . 2 𝑥(∃𝑥𝜑 ∧ ∃*𝑥𝜑)
51, 4nfxfr 1854 1 𝑥∃!𝑥𝜑
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399  ∃wex 1781  Ⅎwnf 1785  ∃*wmo 2620  ∃!weu 2652 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 2145  ax-11 2161  ax-12 2178 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2622  df-eu 2653 This theorem is referenced by: (None)
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