Theorem nfmod 1917

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 1904	. 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
2		nfeud.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfeud.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	2, 3	nfexd 1644	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
5	2, 3	nfeud 1916	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
6	4, 5	nfimd 1477	. 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓))
7	1, 6	nfxfrd 1364	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1349  ∃wex 1381  ∃!weu 1900  ∃*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by:  nfmo  1920