Theorem nfeud 2623

Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2625. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker nfeudw 2622 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfeud.1	⊢ Ⅎ𝑦𝜑
nfeud.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfeud	⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Proof of Theorem nfeud

Step	Hyp	Ref	Expression
1		nfeud.1	. 2 ⊢ Ⅎ𝑦𝜑
2		nfeud.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3	2	adantr 486	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
4	1, 3	nfeud2 2621	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  Ⅎwnf 1816  ∃!weu 2599

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 2179  ax-11 2195  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2570  df-eu 2600

This theorem is used by:  nfeu  2625