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Theorem nf3an 1530
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 949 . 2 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
2 nfan.1 . . . 4 𝑥𝜑
3 nfan.2 . . . 4 𝑥𝜓
42, 3nfan 1529 . . 3 𝑥(𝜑𝜓)
5 nfan.3 . . 3 𝑥𝜒
64, 5nfan 1529 . 2 𝑥((𝜑𝜓) ∧ 𝜒)
71, 6nfxfr 1435 1 𝑥(𝜑𝜓𝜒)
Colors of variables: wff set class
Syntax hints:  wa 103  w3a 947  wnf 1421
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 1408  ax-gen 1410  ax-4 1472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-nf 1422
This theorem is referenced by:  vtocl3gaf  2729  mob  2839  nfop  3691  mkvprop  7000  seq3f1olemstep  10242  seq3f1olemp  10243  nfsum1  11093  nfsum  11094
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