Theorem nfeu1ALT 2592

Description: Alternate proof of nfeu1 2591. 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 1913 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 2572	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		nfe1 2151	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
3		nfmo1 2560	. . 3 ⊢ Ⅎ𝑥∃*𝑥𝜑
4	2, 3	nfan 1898	. 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 ∧ ∃*𝑥𝜑)
5	1, 4	nfxfr 1851	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

Syntax hints:   ∧ wa 395  ∃wex 1777  Ⅎwnf 1781  ∃*wmo 2541  ∃!weu 2571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-mo 2543  df-eu 2572

This theorem is referenced by: (None)