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Theorem bnj523 30211
 Description: Technical lemma for bnj852 30245. 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.)
Hypotheses
Ref Expression
bnj523.1 (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj523.2 (𝜑′[𝑀 / 𝑛]𝜑)
bnj523.3 𝑀 ∈ V
Assertion
Ref Expression
bnj523 (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑛   𝑛,𝐹   𝑅,𝑛   𝑛,𝑋
Allowed substitution hints:   𝜑(𝑛)   𝑀(𝑛)   𝜑′(𝑛)

Proof of Theorem bnj523
StepHypRef Expression
1 bnj523.2 . 2 (𝜑′[𝑀 / 𝑛]𝜑)
2 bnj523.1 . . 3 (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
32sbcbii 3458 . 2 ([𝑀 / 𝑛]𝜑[𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
4 bnj523.3 . . 3 𝑀 ∈ V
54bnj525 30061 . 2 ([𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
61, 3, 53bitri 285 1 (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ∅c0 3874  ‘cfv 5804   predc-bnj14 30007 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  bnj600  30243  bnj908  30255  bnj934  30259
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