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Theorem nf3an 1546
Description: If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
nfan.1 𝑥𝜑
nfan.2 𝑥𝜓
nfan.3 𝑥𝜒
Assertion
Ref Expression
nf3an 𝑥(𝜑𝜓𝜒)

Proof of Theorem nf3an
StepHypRef Expression
1 df-3an 965 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
2 nfan.1 . . . 4 𝑥𝜑
3 nfan.2 . . . 4 𝑥𝜓
42, 3nfan 1545 . . 3 𝑥(𝜑𝜓)
5 nfan.3 . . 3 𝑥𝜒
64, 5nfan 1545 . 2 𝑥((𝜑𝜓) ∧ 𝜒)
71, 6nfxfr 1451 1 𝑥(𝜑𝜓𝜒)
Colors of variables: wff set class
Syntax hints:  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  ax-4 1488
This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438
This theorem is referenced by:  vtocl3gaf  2758  mob  2869  nfop  3728  mkvprop  7039  seq3f1olemstep  10304  seq3f1olemp  10305  nfsum1  11156  nfsum  11157
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