Theorem nfan1 1881

Description: A closed form of nfan 1824. (Contributed by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
nfan1.1	⊢ Ⅎxφ
nfan1.2	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfan1	⊢ Ⅎx(φ ∧ ψ)

Proof of Theorem nfan1

Step	Hyp	Ref	Expression
1		nfan1.2	. . . . 5 ⊢ (φ → Ⅎxψ)
2	1	nfrd 1763	. . . 4 ⊢ (φ → (ψ → ∀xψ))
3	2	imdistani 671	. . 3 ⊢ ((φ ∧ ψ) → (φ ∧ ∀xψ))
4		nfan1.1	. . . 4 ⊢ Ⅎxφ
5	4	19.28 1870	. . 3 ⊢ (∀x(φ ∧ ψ) ↔ (φ ∧ ∀xψ))
6	3, 5	sylibr 203	. 2 ⊢ ((φ ∧ ψ) → ∀x(φ ∧ ψ))
7	6	nfi 1551	1 ⊢ Ⅎx(φ ∧ ψ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544

This proof depends on 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-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is used by:  spimed  1977  ralcom2  2776  sbcralt  3119  sbcrext  3120  csbiebt  3173