Theorem nfeud 2591

Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2593. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker nfeudw 2590 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 2589	1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  Ⅎwnf 1782  ∃!weu 2567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-mo 2539  df-eu 2568

This theorem is referenced by:  nfeu  2593