Theorem nf3and 1501

Description: Deduction form of bound-variable hypothesis builder nf3an 1498. (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 921	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
2		nfand.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		nfand.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4	2, 3	nfand 1500	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒))
5		nfand.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜃)
6	4, 5	nfand 1500	. 2 ⊢ (𝜑 → Ⅎ𝑥((𝜓 ∧ 𝜒) ∧ 𝜃))
7	1, 6	nfxfrd 1404	1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒 ∧ 𝜃))

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919  Ⅎwnf 1389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378

This theorem depends on definitions:  df-bi 115  df-3an 921  df-nf 1390

This theorem is referenced by: (None)