Theorem nfmod2 2217

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
nfeud2.1	⊢ Ⅎyφ
nfeud2.2	⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)

Assertion

Ref	Expression
nfmod2	⊢ (φ → Ⅎx∃*yψ)

Proof of Theorem nfmod2

Step	Hyp	Ref	Expression
1		df-mo 2209	. 2 ⊢ (∃*yψ ↔ (∃yψ → ∃!yψ))
2		nfeud2.1	. . . 4 ⊢ Ⅎyφ
3		nfeud2.2	. . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)
4	2, 3	nfexd2 1973	. . 3 ⊢ (φ → Ⅎx∃yψ)
5	2, 3	nfeud2 2216	. . 3 ⊢ (φ → Ⅎx∃!yψ)
6	4, 5	nfimd 1808	. 2 ⊢ (φ → Ⅎx(∃yψ → ∃!yψ))
7	1, 6	nfxfrd 1571	1 ⊢ (φ → Ⅎx∃*yψ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  ∃!weu 2204  ∃*wmo 2205

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  df-mo 2209

This theorem is referenced by:  nfmod  2219  nfrmod  2785  nfrmo  2787