Theorem nfmod 2094

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 2081	. 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
2		nfeud.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfeud.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	2, 3	nfexd 1807	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
5	2, 3	nfeud 2093	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
6	4, 5	nfimd 1631	. 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓))
7	1, 6	nfxfrd 1521	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1506  ∃wex 1538  ∃!weu 2077  ∃*wmo 2078

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by:  nfmo  2097