Theorem nf3and 1902

Description: Deduction form of bound-variable hypothesis builder nf3an 1905. (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 1087	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
2		nfand.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		nfand.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4	2, 3	nfand 1901	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒))
5		nfand.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜃)
6	4, 5	nfand 1901	. 2 ⊢ (𝜑 → Ⅎ𝑥((𝜓 ∧ 𝜒) ∧ 𝜃))
7	1, 6	nfxfrd 1857	1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒 ∧ 𝜃))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-ex 1784  df-nf 1788

This theorem is referenced by:  nfttrcld  33696