Theorem nf3and 1823

Description: Deduction form of bound-variable hypothesis builder nf3an 1827. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)

Hypotheses

Ref	Expression
nfand.1	⊢ (φ → Ⅎxψ)
nfand.2	⊢ (φ → Ⅎxχ)
nfand.3	⊢ (φ → Ⅎxθ)

Assertion

Ref	Expression
nf3and	⊢ (φ → Ⅎx(ψ ∧ χ ∧ θ))

Proof of Theorem nf3and

Step	Hyp	Ref	Expression
1		df-3an 936	. 2 ⊢ ((ψ ∧ χ ∧ θ) ↔ ((ψ ∧ χ) ∧ θ))
2		nfand.1	. . . 4 ⊢ (φ → Ⅎxψ)
3		nfand.2	. . . 4 ⊢ (φ → Ⅎxχ)
4	2, 3	nfand 1822	. . 3 ⊢ (φ → Ⅎx(ψ ∧ χ))
5		nfand.3	. . 3 ⊢ (φ → Ⅎxθ)
6	4, 5	nfand 1822	. 2 ⊢ (φ → Ⅎx((ψ ∧ χ) ∧ θ))
7	1, 6	nfxfrd 1571	1 ⊢ (φ → Ⅎx(ψ ∧ χ ∧ θ))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  Ⅎwnf 1544

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-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)