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Theorem tfinds2 4843
 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 4339 . . . 4
3 tfinds2.1 . . . . 5
43imbi2d 308 . . . 4
52, 4sbcie 3195 . . 3
61, 5mpbir 201 . 2
7 vex 2959 . . . . . 6
8 tfinds2.5 . . . . . . . 8
98a2d 24 . . . . . . 7
109sbcth 3175 . . . . . 6
117, 10ax-mp 8 . . . . 5
12 sbcimg 3202 . . . . . 6
137, 12ax-mp 8 . . . . 5
1411, 13mpbi 200 . . . 4
15 sbcel1gv 3220 . . . . 5
167, 15ax-mp 8 . . . 4
17 sbcimg 3202 . . . . 5
187, 17ax-mp 8 . . . 4
1914, 16, 183imtr3i 257 . . 3
20 tfinds2.2 . . . . . . 7
2120bicomd 193 . . . . . 6
2221equcoms 1693 . . . . 5
2322imbi2d 308 . . . 4
247, 23sbcie 3195 . . 3
25 vex 2959 . . . . . . 7
2625sucex 4791 . . . . . 6
27 tfinds2.3 . . . . . . 7
2827imbi2d 308 . . . . . 6
2926, 28sbcie 3195 . . . . 5
3029sbcbii 3216 . . . 4
31 suceq 4646 . . . . 5
3231sbcco2 3184 . . . 4
3330, 32bitr3i 243 . . 3
3419, 24, 333imtr3g 261 . 2
35 sbsbc 3165 . . . 4
3623sbralie 2945 . . . 4
3735, 36bitr3i 243 . . 3
38 r19.21v 2793 . . . . . . . 8
39 tfinds2.6 . . . . . . . . 9
4039a2d 24 . . . . . . . 8
4138, 40syl5bi 209 . . . . . . 7
4241sbcth 3175 . . . . . 6
4325, 42ax-mp 8 . . . . 5
44 sbcimg 3202 . . . . . 6
4525, 44ax-mp 8 . . . . 5
4643, 45mpbi 200 . . . 4
47 limeq 4593 . . . . 5
4825, 47sbcie 3195 . . . 4
49 sbcimg 3202 . . . . 5
5025, 49ax-mp 8 . . . 4
5146, 48, 503imtr3i 257 . . 3
5237, 51syl5bir 210 . 2
536, 34, 52tfindes 4842 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652  wsb 1658   wcel 1725  wral 2705  cvv 2956  wsbc 3161  c0 3628  con0 4581   wlim 4582   csuc 4583 This theorem is referenced by:  abianfplem  6715  inar1  8650  grur1a  8694 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587
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