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Theorem bnj1148 34989
Description: Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1148 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)

Proof of Theorem bnj1148
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elisset 2821 . . . . 5 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21adantl 481 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥 𝑥 = 𝑋)
3 bnj93 34856 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
4 eleq1 2827 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
54anbi2d 630 . . . . . 6 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑥𝐴) ↔ (𝑅 FrSe 𝐴𝑋𝐴)))
6 bnj602 34908 . . . . . . 7 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
76eleq1d 2824 . . . . . 6 (𝑥 = 𝑋 → ( pred(𝑥, 𝐴, 𝑅) ∈ V ↔ pred(𝑋, 𝐴, 𝑅) ∈ V))
85, 7imbi12d 344 . . . . 5 (𝑥 = 𝑋 → (((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)))
93, 8mpbii 233 . . . 4 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
102, 9bnj593 34738 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
1110bnj937 34764 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
1211pm2.43i 52 1 ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478   predc-bnj14 34681   FrSe w-bnj15 34685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-bnj14 34682  df-bnj13 34684  df-bnj15 34686
This theorem is referenced by:  bnj1136  34990  bnj1413  35028
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