Theorem nfeu 2220

Description: Bound-variable hypothesis builder for "at most one." Note that x and y needn't be distinct (this makes the proof more difficult). (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypothesis

Ref	Expression
nfeu.1	⊢ Ⅎxφ

Assertion

Ref	Expression
nfeu	⊢ Ⅎx∃!yφ

Proof of Theorem nfeu

Step	Hyp	Ref	Expression
1		nftru 1554	. . 3 ⊢ Ⅎy ⊤
2		nfeu.1	. . . 4 ⊢ Ⅎxφ
3	2	a1i 10	. . 3 ⊢ ( ⊤ → Ⅎxφ)
4	1, 3	nfeud 2218	. 2 ⊢ ( ⊤ → Ⅎx∃!yφ)
5	4	trud 1323	1 ⊢ Ⅎx∃!yφ

Syntax hints:   ⊤ wtru 1316  Ⅎwnf 1544  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  2eu7  2290  2eu8  2291