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Theorem nf3and 1461
 Description: Deduction form of bound-variable hypothesis builder nf3an 1458. (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(𝜓𝜒𝜃))

Proof of Theorem nf3and
StepHypRef Expression
1 df-3an 887 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∧ 𝜃))
2 nfand.1 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
3 nfand.2 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
42, 3nfand 1460 . . 3 (𝜑 → Ⅎ𝑥(𝜓𝜒))
5 nfand.3 . . 3 (𝜑 → Ⅎ𝑥𝜃)
64, 5nfand 1460 . 2 (𝜑 → Ⅎ𝑥((𝜓𝜒) ∧ 𝜃))
71, 6nfxfrd 1364 1 (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885  Ⅎwnf 1349 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338 This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350 This theorem is referenced by: (None)
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