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Theorem bj-findis 10104
 Description: Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 10072 for a bounded version not requiring ax-setind 4262. See finds 4323 for a proof in IZF. From this version, it is easy to prove of finds 4323, finds2 4324, finds1 4325. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-findis.nf0 𝑥𝜓
bj-findis.nf1 𝑥𝜒
bj-findis.nfsuc 𝑥𝜃
bj-findis.0 (𝑥 = ∅ → (𝜓𝜑))
bj-findis.1 (𝑥 = 𝑦 → (𝜑𝜒))
bj-findis.suc (𝑥 = suc 𝑦 → (𝜃𝜑))
Assertion
Ref Expression
bj-findis ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝜃(𝑥,𝑦)

Proof of Theorem bj-findis
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bj-nn0suc 10089 . . . . 5 (𝑧 ∈ ω ↔ (𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦))
2 pm3.21 251 . . . . . . . 8 (𝜓 → (𝑧 = ∅ → (𝑧 = ∅ ∧ 𝜓)))
32ad2antrr 457 . . . . . . 7 (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) ∧ ∀𝑦𝑧 (𝑦 ∈ ω → 𝜒)) → (𝑧 = ∅ → (𝑧 = ∅ ∧ 𝜓)))
4 pm2.04 76 . . . . . . . . . . 11 ((𝑦𝑧 → (𝑦 ∈ ω → 𝜒)) → (𝑦 ∈ ω → (𝑦𝑧𝜒)))
54ralimi2 2381 . . . . . . . . . 10 (∀𝑦𝑧 (𝑦 ∈ ω → 𝜒) → ∀𝑦 ∈ ω (𝑦𝑧𝜒))
6 imim2 49 . . . . . . . . . . . 12 ((𝜒𝜃) → ((𝑦𝑧𝜒) → (𝑦𝑧𝜃)))
76ral2imi 2385 . . . . . . . . . . 11 (∀𝑦 ∈ ω (𝜒𝜃) → (∀𝑦 ∈ ω (𝑦𝑧𝜒) → ∀𝑦 ∈ ω (𝑦𝑧𝜃)))
87imp 115 . . . . . . . . . 10 ((∀𝑦 ∈ ω (𝜒𝜃) ∧ ∀𝑦 ∈ ω (𝑦𝑧𝜒)) → ∀𝑦 ∈ ω (𝑦𝑧𝜃))
95, 8sylan2 270 . . . . . . . . 9 ((∀𝑦 ∈ ω (𝜒𝜃) ∧ ∀𝑦𝑧 (𝑦 ∈ ω → 𝜒)) → ∀𝑦 ∈ ω (𝑦𝑧𝜃))
10 r19.29 2450 . . . . . . . . . . 11 ((∀𝑦 ∈ ω (𝑦𝑧𝜃) ∧ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ∃𝑦 ∈ ω ((𝑦𝑧𝜃) ∧ 𝑧 = suc 𝑦))
11 vex 2560 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
1211sucid 4154 . . . . . . . . . . . . . . 15 𝑦 ∈ suc 𝑦
13 eleq2 2101 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑦 → (𝑦𝑧𝑦 ∈ suc 𝑦))
1412, 13mpbiri 157 . . . . . . . . . . . . . 14 (𝑧 = suc 𝑦𝑦𝑧)
15 ax-1 5 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑦 → ((𝑦𝑧𝜃) → 𝑧 = suc 𝑦))
16 pm2.27 35 . . . . . . . . . . . . . . 15 (𝑦𝑧 → ((𝑦𝑧𝜃) → 𝜃))
1715, 16anim12ii 325 . . . . . . . . . . . . . 14 ((𝑧 = suc 𝑦𝑦𝑧) → ((𝑦𝑧𝜃) → (𝑧 = suc 𝑦𝜃)))
1814, 17mpdan 398 . . . . . . . . . . . . 13 (𝑧 = suc 𝑦 → ((𝑦𝑧𝜃) → (𝑧 = suc 𝑦𝜃)))
1918impcom 116 . . . . . . . . . . . 12 (((𝑦𝑧𝜃) ∧ 𝑧 = suc 𝑦) → (𝑧 = suc 𝑦𝜃))
2019reximi 2416 . . . . . . . . . . 11 (∃𝑦 ∈ ω ((𝑦𝑧𝜃) ∧ 𝑧 = suc 𝑦) → ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃))
2110, 20syl 14 . . . . . . . . . 10 ((∀𝑦 ∈ ω (𝑦𝑧𝜃) ∧ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃))
2221ex 108 . . . . . . . . 9 (∀𝑦 ∈ ω (𝑦𝑧𝜃) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))
239, 22syl 14 . . . . . . . 8 ((∀𝑦 ∈ ω (𝜒𝜃) ∧ ∀𝑦𝑧 (𝑦 ∈ ω → 𝜒)) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))
2423adantll 445 . . . . . . 7 (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) ∧ ∀𝑦𝑧 (𝑦 ∈ ω → 𝜒)) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))
253, 24orim12d 700 . . . . . 6 (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) ∧ ∀𝑦𝑧 (𝑦 ∈ ω → 𝜒)) → ((𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃))))
2625ex 108 . . . . 5 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (∀𝑦𝑧 (𝑦 ∈ ω → 𝜒) → ((𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))))
271, 26syl7bi 154 . . . 4 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (∀𝑦𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))))
2827alrimiv 1754 . . 3 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑧(∀𝑦𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))))
29 nfv 1421 . . . . 5 𝑥 𝑦 ∈ ω
30 bj-findis.nf1 . . . . 5 𝑥𝜒
3129, 30nfim 1464 . . . 4 𝑥(𝑦 ∈ ω → 𝜒)
32 nfv 1421 . . . . 5 𝑥 𝑧 ∈ ω
33 nfv 1421 . . . . . . 7 𝑥 𝑧 = ∅
34 bj-findis.nf0 . . . . . . 7 𝑥𝜓
3533, 34nfan 1457 . . . . . 6 𝑥(𝑧 = ∅ ∧ 𝜓)
36 nfcv 2178 . . . . . . 7 𝑥ω
37 nfv 1421 . . . . . . . 8 𝑥 𝑧 = suc 𝑦
38 bj-findis.nfsuc . . . . . . . 8 𝑥𝜃
3937, 38nfan 1457 . . . . . . 7 𝑥(𝑧 = suc 𝑦𝜃)
4036, 39nfrexxy 2361 . . . . . 6 𝑥𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)
4135, 40nfor 1466 . . . . 5 𝑥((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃))
4232, 41nfim 1464 . . . 4 𝑥(𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))
43 nfv 1421 . . . 4 𝑧(𝑥 ∈ ω → 𝜑)
44 nfv 1421 . . . 4 𝑧(𝑦 ∈ ω → 𝜒)
45 eleq1 2100 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
4645biimprd 147 . . . . 5 (𝑥 = 𝑦 → (𝑦 ∈ ω → 𝑥 ∈ ω))
47 bj-findis.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
4846, 47imim12d 68 . . . 4 (𝑥 = 𝑦 → ((𝑥 ∈ ω → 𝜑) → (𝑦 ∈ ω → 𝜒)))
49 eleq1 2100 . . . . . 6 (𝑥 = 𝑧 → (𝑥 ∈ ω ↔ 𝑧 ∈ ω))
5049biimpd 132 . . . . 5 (𝑥 = 𝑧 → (𝑥 ∈ ω → 𝑧 ∈ ω))
51 eqtr 2057 . . . . . . . 8 ((𝑥 = 𝑧𝑧 = ∅) → 𝑥 = ∅)
52 bj-findis.0 . . . . . . . 8 (𝑥 = ∅ → (𝜓𝜑))
5351, 52syl 14 . . . . . . 7 ((𝑥 = 𝑧𝑧 = ∅) → (𝜓𝜑))
5453expimpd 345 . . . . . 6 (𝑥 = 𝑧 → ((𝑧 = ∅ ∧ 𝜓) → 𝜑))
55 eqtr 2057 . . . . . . . . 9 ((𝑥 = 𝑧𝑧 = suc 𝑦) → 𝑥 = suc 𝑦)
56 bj-findis.suc . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝜃𝜑))
5755, 56syl 14 . . . . . . . 8 ((𝑥 = 𝑧𝑧 = suc 𝑦) → (𝜃𝜑))
5857expimpd 345 . . . . . . 7 (𝑥 = 𝑧 → ((𝑧 = suc 𝑦𝜃) → 𝜑))
5958rexlimdvw 2436 . . . . . 6 (𝑥 = 𝑧 → (∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃) → 𝜑))
6054, 59jaod 637 . . . . 5 (𝑥 = 𝑧 → (((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)) → 𝜑))
6150, 60imim12d 68 . . . 4 (𝑥 = 𝑧 → ((𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃))) → (𝑥 ∈ ω → 𝜑)))
6231, 42, 43, 44, 48, 61setindis 10092 . . 3 (∀𝑧(∀𝑦𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))) → ∀𝑥(𝑥 ∈ ω → 𝜑))
6328, 62syl 14 . 2 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥(𝑥 ∈ ω → 𝜑))
64 df-ral 2311 . 2 (∀𝑥 ∈ ω 𝜑 ↔ ∀𝑥(𝑥 ∈ ω → 𝜑))
6563, 64sylibr 137 1 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629  ∀wal 1241   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  ∅c0 3224  suc csuc 4102  ωcom 4313 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-bd0 9933  ax-bdim 9934  ax-bdan 9935  ax-bdor 9936  ax-bdn 9937  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051 This theorem is referenced by:  bj-findisg  10105  bj-findes  10106
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