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Theorem bnj526 32047
Description: Technical lemma for bnj852 32080. 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
bnj526.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj526.2 (𝜑″[𝐺 / 𝑓]𝜑)
bnj526.3 𝐺 ∈ V
Assertion
Ref Expression
bnj526 (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑅,𝑓   𝑓,𝑋
Allowed substitution hints:   𝜑(𝑓)   𝜑″(𝑓)

Proof of Theorem bnj526
StepHypRef Expression
1 bnj526.2 . 2 (𝜑″[𝐺 / 𝑓]𝜑)
2 bnj526.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
32sbcbii 3832 . 2 ([𝐺 / 𝑓]𝜑[𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
4 bnj526.3 . . 3 𝐺 ∈ V
5 fveq1 6665 . . . 4 (𝑓 = 𝐺 → (𝑓‘∅) = (𝐺‘∅))
65eqeq1d 2826 . . 3 (𝑓 = 𝐺 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)))
74, 6sbcie 3815 . 2 ([𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
81, 3, 73bitri 298 1 (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1530  wcel 2106  Vcvv 3499  [wsbc 3775  c0 4294  cfv 6351   predc-bnj14 31845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-rex 3148  df-v 3501  df-sbc 3776  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359
This theorem is referenced by:  bnj607  32075
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