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Theorem bnj1148 30769
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 3201 . . . . 5 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21adantl 482 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥 𝑥 = 𝑋)
3 bnj93 30638 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
4 eleq1 2686 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
54anbi2d 739 . . . . . 6 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑥𝐴) ↔ (𝑅 FrSe 𝐴𝑋𝐴)))
6 bnj602 30690 . . . . . . 7 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
76eleq1d 2683 . . . . . 6 (𝑥 = 𝑋 → ( pred(𝑥, 𝐴, 𝑅) ∈ V ↔ pred(𝑋, 𝐴, 𝑅) ∈ V))
85, 7imbi12d 334 . . . . 5 (𝑥 = 𝑋 → (((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)))
93, 8mpbii 223 . . . 4 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
102, 9bnj593 30520 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
1110bnj937 30547 . 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 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186   predc-bnj14 30458   FrSe w-bnj15 30462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-bnj14 30459  df-bnj13 30461  df-bnj15 30463
This theorem is referenced by:  bnj1136  30770  bnj1413  30808
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