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Theorem nf3and 1895
Description: Deduction form of bound-variable hypothesis builder nf3an 1898. (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 1085 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∧ 𝜃))
2 nfand.1 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
3 nfand.2 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
42, 3nfand 1894 . . 3 (𝜑 → Ⅎ𝑥(𝜓𝜒))
5 nfand.3 . . 3 (𝜑 → Ⅎ𝑥𝜃)
64, 5nfand 1894 . 2 (𝜑 → Ⅎ𝑥((𝜓𝜒) ∧ 𝜃))
71, 6nfxfrd 1850 1 (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083  wnf 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-ex 1777  df-nf 1781
This theorem is referenced by: (None)
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