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Theorem bnj93 30918
Description: Technical lemma for bnj97 30921. 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 30744 . . . 4 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
21simprbi 480 . . 3 (𝑅 FrSe 𝐴𝑅 Se 𝐴)
3 df-bnj13 30742 . . 3 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
42, 3sylib 208 . 2 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
54r19.21bi 2931 1 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1989  wral 2911  Vcvv 3198   Fr wfr 5068   predc-bnj14 30739   Se w-bnj13 30741   FrSe w-bnj15 30743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-ral 2916  df-bnj13 30742  df-bnj15 30744
This theorem is referenced by:  bnj96  30920  bnj97  30921  bnj149  30930  bnj150  30931  bnj518  30941  bnj1148  31049
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