Theorem nfeud 2595

Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2597. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker nfeudw 2594 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 480	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
4	1, 3	nfeud2 2593	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  Ⅎwnf 1781  ∃!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  ax-13 2380

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:  nfeu  2597