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Theorem nfeu1ALT 2608
Description: Alternate proof of nfeu1 2607. 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 1873 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 2583 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2 nfe1 2087 . . 3 𝑥𝑥𝜑
3 nfmo1 2569 . . 3 𝑥∃*𝑥𝜑
42, 3nfan 1862 . 2 𝑥(∃𝑥𝜑 ∧ ∃*𝑥𝜑)
51, 4nfxfr 1815 1 𝑥∃!𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wa 387  wex 1742  wnf 1746  ∃*wmo 2545  ∃!weu 2582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-11 2093  ax-12 2106
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-mo 2547  df-eu 2583
This theorem is referenced by: (None)
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