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Theorem bnj523 31473
Description: Technical lemma for bnj852 31507. 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 3690 . 2 ([𝑀 / 𝑛]𝜑[𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
4 bnj523.3 . . 3 𝑀 ∈ V
54bnj525 31324 . 2 ([𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
61, 3, 53bitri 289 1 (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  wcel 2157  Vcvv 3386  [wsbc 3634  c0 4116  cfv 6102   predc-bnj14 31273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-v 3388  df-sbc 3635
This theorem is referenced by:  bnj600  31505  bnj908  31517  bnj934  31521
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