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Theorem bnj523 34901
Description: Technical lemma for bnj852 34935. 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 3846 . 2 ([𝑀 / 𝑛]𝜑[𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
4 bnj523.3 . . 3 𝑀 ∈ V
54bnj525 34752 . 2 ([𝑀 / 𝑛](𝐹‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
61, 3, 53bitri 297 1 (𝜑′ ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  [wsbc 3788  c0 4333  cfv 6561   predc-bnj14 34702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789
This theorem is referenced by:  bnj600  34933  bnj908  34945  bnj934  34949
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