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Theorem bnj93 31707
Description: Technical lemma for bnj97 31710. 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 31536 . . . 4 (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Se 𝐴))
21simprbi 497 . . 3 (𝑅 FrSe 𝐴𝑅 Se 𝐴)
3 df-bnj13 31534 . . 3 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
42, 3sylib 219 . 2 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
54r19.21bi 3173 1 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2079  wral 3103  Vcvv 3432   Fr wfr 5391   predc-bnj14 31531   Se w-bnj13 31533   FrSe w-bnj15 31535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-12 2139
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1760  df-ral 3108  df-bnj13 31534  df-bnj15 31536
This theorem is referenced by:  bnj96  31709  bnj97  31710  bnj149  31719  bnj150  31720  bnj518  31730  bnj1148  31838
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