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Theorem nf3and 1549
Description: Deduction form of bound-variable hypothesis builder nf3an 1546. (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 965 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∧ 𝜃))
2 nfand.1 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
3 nfand.2 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
42, 3nfand 1548 . . 3 (𝜑 → Ⅎ𝑥(𝜓𝜒))
5 nfand.3 . . 3 (𝜑 → Ⅎ𝑥𝜃)
64, 5nfand 1548 . 2 (𝜑 → Ⅎ𝑥((𝜓𝜒) ∧ 𝜃))
71, 6nfxfrd 1452 1 (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963  wnf 1437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426
This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438
This theorem is referenced by: (None)
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