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Theorem bnj517 28917
 Description: Technical lemma for bnj518 28918. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj517.1
bnj517.2
Assertion
Ref Expression
bnj517
Distinct variable groups:   ,,,   ,,   ,,
Allowed substitution hints:   (,,)   (,,)   (,,)   ()   ()   (,,)

Proof of Theorem bnj517
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bnj213 28914 . . . . 5
2 fveq2 5525 . . . . . . 7
3 simpl2 959 . . . . . . . 8
4 bnj517.1 . . . . . . . 8
53, 4sylib 188 . . . . . . 7
62, 5sylan9eqr 2337 . . . . . 6
76sseq1d 3205 . . . . 5
81, 7mpbiri 224 . . . 4
9 bnj517.2 . . . . . . 7
10 r19.29r 2684 . . . . . . . . . 10
11 eleq1 2343 . . . . . . . . . . . . . 14
1211biimpd 198 . . . . . . . . . . . . 13
13 fveq2 5525 . . . . . . . . . . . . . . 15
1413eqeq1d 2291 . . . . . . . . . . . . . 14
15 bnj213 28914 . . . . . . . . . . . . . . . . 17
1615rgenw 2610 . . . . . . . . . . . . . . . 16
17 iunss 3943 . . . . . . . . . . . . . . . 16
1816, 17mpbir 200 . . . . . . . . . . . . . . 15
19 sseq1 3199 . . . . . . . . . . . . . . 15
2018, 19mpbiri 224 . . . . . . . . . . . . . 14
2114, 20syl6bir 220 . . . . . . . . . . . . 13
2212, 21imim12d 68 . . . . . . . . . . . 12
2322imp 418 . . . . . . . . . . 11
2423rexlimivw 2663 . . . . . . . . . 10
2510, 24syl 15 . . . . . . . . 9
2625ex 423 . . . . . . . 8
2726com3l 75 . . . . . . 7
289, 27sylbi 187 . . . . . 6
29283ad2ant3 978 . . . . 5
3029imp31 421 . . . 4
31 simpr 447 . . . . . 6
32 simpl1 958 . . . . . 6
33 elnn 4666 . . . . . 6
3431, 32, 33syl2anc 642 . . . . 5
35 nn0suc 4680 . . . . 5
3634, 35syl 15 . . . 4
378, 30, 36mpjaodan 761 . . 3
3837ralrimiva 2626 . 2
39 fveq2 5525 . . . 4
4039sseq1d 3205 . . 3
4140cbvralv 2764 . 2
4238, 41sylib 188 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wo 357   wa 358   w3a 934   wceq 1623   wcel 1684  wral 2543  wrex 2544   wss 3152  c0 3455  ciun 3905   csuc 4394  com 4656  cfv 5255   c-bnj14 28713 This theorem is referenced by:  bnj518  28918 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-iota 5219  df-fv 5263  df-bnj14 28714
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