Theorem nfmod 2036

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 2023	. 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
2		nfeud.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfeud.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	2, 3	nfexd 1754	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
5	2, 3	nfeud 2035	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
6	4, 5	nfimd 1578	. 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓))
7	1, 6	nfxfrd 1468	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1453  ∃wex 1485  ∃!weu 2019  ∃*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by:  nfmo  2039