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Theorem bnj1148 31402
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 3367 . . . . 5 (𝑋𝐴 → ∃𝑥 𝑥 = 𝑋)
21adantl 467 . . . 4 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥 𝑥 = 𝑋)
3 bnj93 31271 . . . . 5 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
4 eleq1 2838 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
54anbi2d 614 . . . . . 6 (𝑥 = 𝑋 → ((𝑅 FrSe 𝐴𝑥𝐴) ↔ (𝑅 FrSe 𝐴𝑋𝐴)))
6 bnj602 31323 . . . . . . 7 (𝑥 = 𝑋 → pred(𝑥, 𝐴, 𝑅) = pred(𝑋, 𝐴, 𝑅))
76eleq1d 2835 . . . . . 6 (𝑥 = 𝑋 → ( pred(𝑥, 𝐴, 𝑅) ∈ V ↔ pred(𝑋, 𝐴, 𝑅) ∈ V))
85, 7imbi12d 333 . . . . 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 31153 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∃𝑥((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V))
1110bnj937 31180 . 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 382   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351   predc-bnj14 31094   FrSe w-bnj15 31098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-bnj14 31095  df-bnj13 31097  df-bnj15 31099
This theorem is referenced by:  bnj1136  31403  bnj1413  31441
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