Theorem nfeu1ALT 2671

Description: Alternate proof of nfeu1 2670. 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 1911 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	⊢ Ⅎ𝑥∃!𝑥𝜑

Proof of Theorem nfeu1ALT

Step	Hyp	Ref	Expression
1		df-eu 2650	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		nfe1 2150	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfmo1 2637	. . 3 ⊢ Ⅎ𝑥∃*𝑥𝜑
4	2, 3	nfan 1896	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 ∧ ∃*𝑥𝜑)
5	1, 4	nfxfr 1849	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

Syntax hints:   ∧ wa 398  ∃wex 1776  Ⅎwnf 1780  ∃*wmo 2616  ∃!weu 2649

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 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2156  ax-12 2172

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-mo 2618  df-eu 2650

This theorem is referenced by: (None)