Theorem nfeu1ALT 2614

Description: Alternate version of nfeu1 2615 with a shorter proof but using ax-12 2213. Bound-variable hypothesis builder for uniqueness. See also nfeu1 2615. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nfeu1ALT	⊢ Ⅎ𝑥∃!𝑥𝜑

Proof of Theorem nfeu1ALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eu6 2600	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		nfexa2 2212	. 2 ⊢ Ⅎ𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
3	1, 2	nfxfr 1886	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816  ∃!weu 2594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-mo 2565  df-eu 2595

This theorem is used by: (None)