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Theorem bnj1148 34002
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 2815 . . . . 5 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21adantl 482 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥 𝑥 = 𝑋)
3 bnj93 33869 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
4 eleq1 2821 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
54anbi2d 629 . . . . . 6 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑥𝐴) ↔ (𝑅 FrSe 𝐴𝑋𝐴)))
6 bnj602 33921 . . . . . . 7 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
76eleq1d 2818 . . . . . 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 232 . . . 4 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
102, 9bnj593 33751 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
1110bnj937 33777 . 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 396   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474   predc-bnj14 33694   FrSe w-bnj15 33698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-bnj14 33695  df-bnj13 33697  df-bnj15 33699
This theorem is referenced by:  bnj1136  34003  bnj1413  34041
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