Theorem nfmod 2097

Description: Bound-variable hypothesis builder for "at most one". (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfmod	⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Proof of Theorem nfmod

Step	Hyp	Ref	Expression
1		df-mo 2084	. 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
2		nfeud.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfeud.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	2, 3	nfexd 1810	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
5	2, 3	nfeud 2096	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
6	4, 5	nfimd 1634	. 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓))
7	1, 6	nfxfrd 1524	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1509  ∃wex 1541  ∃!weu 2080  ∃*wmo 2081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084

This theorem is referenced by:  nfmo  2100