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Theorem nfeu1ALT 2587
Description: Alternate proof of nfeu1 2586. 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 1912 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 2567 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2 nfe1 2148 . . 3 𝑥𝑥𝜑
3 nfmo1 2555 . . 3 𝑥∃*𝑥𝜑
42, 3nfan 1897 . 2 𝑥(∃𝑥𝜑 ∧ ∃*𝑥𝜑)
51, 4nfxfr 1850 1 𝑥∃!𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1776  wnf 1780  ∃*wmo 2536  ∃!weu 2566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-mo 2538  df-eu 2567
This theorem is referenced by: (None)
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