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Theorem bnj93 34494
Description: Technical lemma for bnj97 34497. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj93 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem bnj93
StepHypRef Expression
1 df-bnj15 34324 . . . 4 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
21simprbi 496 . . 3 (𝑅 FrSe 𝐴𝑅 Se 𝐴)
3 df-bnj13 34322 . . 3 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
42, 3sylib 217 . 2 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
54r19.21bi 3245 1 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  wral 3058  Vcvv 3471   Fr wfr 5630   predc-bnj14 34319   Se w-bnj13 34321   FrSe w-bnj15 34323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2167
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-ral 3059  df-bnj13 34322  df-bnj15 34324
This theorem is referenced by:  bnj96  34496  bnj97  34497  bnj149  34506  bnj150  34507  bnj518  34517  bnj1148  34627
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