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Theorem tfinds2 4545
 Description: Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)
Hypotheses
Ref Expression
tfinds2.1
tfinds2.2
tfinds2.3
tfinds2.4
tfinds2.5
tfinds2.6
Assertion
Ref Expression
tfinds2
Distinct variable groups:   ,,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem tfinds2
StepHypRef Expression
1 tfinds2.4 . . 3
2 0ex 4047 . . . 4
3 tfinds2.1 . . . . 5
43imbi2d 309 . . . 4
52, 4sbcie 2955 . . 3
61, 5mpbir 202 . 2
7 vex 2730 . . . . . 6
8 tfinds2.5 . . . . . . . 8
98a2d 25 . . . . . . 7
109sbcth 2935 . . . . . 6
117, 10ax-mp 10 . . . . 5
12 sbcimg 2962 . . . . . 6
137, 12ax-mp 10 . . . . 5
1411, 13mpbi 201 . . . 4
15 sbcel1gv 2980 . . . . 5
167, 15ax-mp 10 . . . 4
17 sbcimg 2962 . . . . 5
187, 17ax-mp 10 . . . 4
1914, 16, 183imtr3i 258 . . 3
20 tfinds2.2 . . . . . . 7
2120bicomd 194 . . . . . 6
2221equcoms 1825 . . . . 5
2322imbi2d 309 . . . 4
247, 23sbcie 2955 . . 3
25 vex 2730 . . . . . . 7
2625sucex 4493 . . . . . 6
27 tfinds2.3 . . . . . . 7
2827imbi2d 309 . . . . . 6
2926, 28sbcie 2955 . . . . 5
3029sbcbii 2976 . . . 4
31 suceq 4350 . . . . 5
3231sbcco2 2944 . . . 4
3330, 32bitr3i 244 . . 3
3419, 24, 333imtr3g 262 . 2
35 sbsbc 2925 . . . 4
3623sbralie 2716 . . . 4
3735, 36bitr3i 244 . . 3
38 r19.21v 2592 . . . . . . . 8
39 tfinds2.6 . . . . . . . . 9
4039a2d 25 . . . . . . . 8
4138, 40syl5bi 210 . . . . . . 7
4241sbcth 2935 . . . . . 6
4325, 42ax-mp 10 . . . . 5
44 sbcimg 2962 . . . . . 6
4525, 44ax-mp 10 . . . . 5
4643, 45mpbi 201 . . . 4
47 limeq 4297 . . . . 5
4825, 47sbcie 2955 . . . 4
49 sbcimg 2962 . . . . 5
5025, 49ax-mp 10 . . . 4
5146, 48, 503imtr3i 258 . . 3
5237, 51syl5bir 211 . 2
536, 34, 52tfindes 4544 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wceq 1619   wcel 1621  wsb 1882  wral 2509  cvv 2727  wsbc 2921  c0 3362  con0 4285   wlim 4286   csuc 4287 This theorem is referenced by:  abianfplem  6356  inar1  8277  grur1a  8321 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291
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