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Theorem bj-findis 14734
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 14702 for a bounded version not requiring ax-setind 4537. See finds 4600 for a proof in IZF. From this version, it is easy to prove of finds 4600, finds2 4601, finds1 4602. (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 14719 . . . . 5 (𝑧 ∈ ω ↔ (𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦))
2 pm3.21 264 . . . . . . . 8 (𝜓 → (𝑧 = ∅ → (𝑧 = ∅ ∧ 𝜓)))
32ad2antrr 488 . . . . . . 7 (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) ∧ ∀𝑦𝑧 (𝑦 ∈ ω → 𝜒)) → (𝑧 = ∅ → (𝑧 = ∅ ∧ 𝜓)))
4 pm2.04 82 . . . . . . . . . . 11 ((𝑦𝑧 → (𝑦 ∈ ω → 𝜒)) → (𝑦 ∈ ω → (𝑦𝑧𝜒)))
54ralimi2 2537 . . . . . . . . . 10 (∀𝑦𝑧 (𝑦 ∈ ω → 𝜒) → ∀𝑦 ∈ ω (𝑦𝑧𝜒))
6 imim2 55 . . . . . . . . . . . 12 ((𝜒𝜃) → ((𝑦𝑧𝜒) → (𝑦𝑧𝜃)))
76ral2imi 2542 . . . . . . . . . . 11 (∀𝑦 ∈ ω (𝜒𝜃) → (∀𝑦 ∈ ω (𝑦𝑧𝜒) → ∀𝑦 ∈ ω (𝑦𝑧𝜃)))
87imp 124 . . . . . . . . . 10 ((∀𝑦 ∈ ω (𝜒𝜃) ∧ ∀𝑦 ∈ ω (𝑦𝑧𝜒)) → ∀𝑦 ∈ ω (𝑦𝑧𝜃))
95, 8sylan2 286 . . . . . . . . 9 ((∀𝑦 ∈ ω (𝜒𝜃) ∧ ∀𝑦𝑧 (𝑦 ∈ ω → 𝜒)) → ∀𝑦 ∈ ω (𝑦𝑧𝜃))
10 r19.29 2614 . . . . . . . . . . 11 ((∀𝑦 ∈ ω (𝑦𝑧𝜃) ∧ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ∃𝑦 ∈ ω ((𝑦𝑧𝜃) ∧ 𝑧 = suc 𝑦))
11 vex 2741 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
1211sucid 4418 . . . . . . . . . . . . . . 15 𝑦 ∈ suc 𝑦
13 eleq2 2241 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑦 → (𝑦𝑧𝑦 ∈ suc 𝑦))
1412, 13mpbiri 168 . . . . . . . . . . . . . 14 (𝑧 = suc 𝑦𝑦𝑧)
15 ax-1 6 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑦 → ((𝑦𝑧𝜃) → 𝑧 = suc 𝑦))
16 pm2.27 40 . . . . . . . . . . . . . . 15 (𝑦𝑧 → ((𝑦𝑧𝜃) → 𝜃))
1715, 16anim12ii 343 . . . . . . . . . . . . . 14 ((𝑧 = suc 𝑦𝑦𝑧) → ((𝑦𝑧𝜃) → (𝑧 = suc 𝑦𝜃)))
1814, 17mpdan 421 . . . . . . . . . . . . 13 (𝑧 = suc 𝑦 → ((𝑦𝑧𝜃) → (𝑧 = suc 𝑦𝜃)))
1918impcom 125 . . . . . . . . . . . 12 (((𝑦𝑧𝜃) ∧ 𝑧 = suc 𝑦) → (𝑧 = suc 𝑦𝜃))
2019reximi 2574 . . . . . . . . . . 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 115 . . . . . . . . 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 476 . . . . . . 7 (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) ∧ ∀𝑦𝑧 (𝑦 ∈ ω → 𝜒)) → (∃𝑦 ∈ ω 𝑧 = suc 𝑦 → ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))
253, 24orim12d 786 . . . . . 6 (((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) ∧ ∀𝑦𝑧 (𝑦 ∈ ω → 𝜒)) → ((𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃))))
2625ex 115 . . . . 5 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (∀𝑦𝑧 (𝑦 ∈ ω → 𝜒) → ((𝑧 = ∅ ∨ ∃𝑦 ∈ ω 𝑧 = suc 𝑦) → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))))
271, 26syl7bi 165 . . . 4 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → (∀𝑦𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))))
2827alrimiv 1874 . . 3 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑧(∀𝑦𝑧 (𝑦 ∈ ω → 𝜒) → (𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))))
29 nfv 1528 . . . . 5 𝑥 𝑦 ∈ ω
30 bj-findis.nf1 . . . . 5 𝑥𝜒
3129, 30nfim 1572 . . . 4 𝑥(𝑦 ∈ ω → 𝜒)
32 nfv 1528 . . . . 5 𝑥 𝑧 ∈ ω
33 nfv 1528 . . . . . . 7 𝑥 𝑧 = ∅
34 bj-findis.nf0 . . . . . . 7 𝑥𝜓
3533, 34nfan 1565 . . . . . 6 𝑥(𝑧 = ∅ ∧ 𝜓)
36 nfcv 2319 . . . . . . 7 𝑥ω
37 nfv 1528 . . . . . . . 8 𝑥 𝑧 = suc 𝑦
38 bj-findis.nfsuc . . . . . . . 8 𝑥𝜃
3937, 38nfan 1565 . . . . . . 7 𝑥(𝑧 = suc 𝑦𝜃)
4036, 39nfrexxy 2516 . . . . . 6 𝑥𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)
4135, 40nfor 1574 . . . . 5 𝑥((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃))
4232, 41nfim 1572 . . . 4 𝑥(𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)))
43 nfv 1528 . . . 4 𝑧(𝑥 ∈ ω → 𝜑)
44 nfv 1528 . . . 4 𝑧(𝑦 ∈ ω → 𝜒)
45 eleq1 2240 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
4645biimprd 158 . . . . 5 (𝑥 = 𝑦 → (𝑦 ∈ ω → 𝑥 ∈ ω))
47 bj-findis.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜒))
4846, 47imim12d 74 . . . 4 (𝑥 = 𝑦 → ((𝑥 ∈ ω → 𝜑) → (𝑦 ∈ ω → 𝜒)))
49 eleq1 2240 . . . . . 6 (𝑥 = 𝑧 → (𝑥 ∈ ω ↔ 𝑧 ∈ ω))
5049biimpd 144 . . . . 5 (𝑥 = 𝑧 → (𝑥 ∈ ω → 𝑧 ∈ ω))
51 eqtr 2195 . . . . . . . 8 ((𝑥 = 𝑧𝑧 = ∅) → 𝑥 = ∅)
52 bj-findis.0 . . . . . . . 8 (𝑥 = ∅ → (𝜓𝜑))
5351, 52syl 14 . . . . . . 7 ((𝑥 = 𝑧𝑧 = ∅) → (𝜓𝜑))
5453expimpd 363 . . . . . 6 (𝑥 = 𝑧 → ((𝑧 = ∅ ∧ 𝜓) → 𝜑))
55 eqtr 2195 . . . . . . . . 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 363 . . . . . . 7 (𝑥 = 𝑧 → ((𝑧 = suc 𝑦𝜃) → 𝜑))
5958rexlimdvw 2598 . . . . . 6 (𝑥 = 𝑧 → (∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃) → 𝜑))
6054, 59jaod 717 . . . . 5 (𝑥 = 𝑧 → (((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃)) → 𝜑))
6150, 60imim12d 74 . . . 4 (𝑥 = 𝑧 → ((𝑧 ∈ ω → ((𝑧 = ∅ ∧ 𝜓) ∨ ∃𝑦 ∈ ω (𝑧 = suc 𝑦𝜃))) → (𝑥 ∈ ω → 𝜑)))
6231, 42, 43, 44, 48, 61setindis 14722 . . 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 2460 . 2 (∀𝑥 ∈ ω 𝜑 ↔ ∀𝑥(𝑥 ∈ ω → 𝜑))
6563, 64sylibr 134 1 ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒𝜃)) → ∀𝑥 ∈ ω 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708  wal 1351   = wceq 1353  wnf 1460  wcel 2148  wral 2455  wrex 2456  c0 3423  suc csuc 4366  ωcom 4590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4130  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-bd0 14568  ax-bdim 14569  ax-bdan 14570  ax-bdor 14571  ax-bdn 14572  ax-bdal 14573  ax-bdex 14574  ax-bdeq 14575  ax-bdel 14576  ax-bdsb 14577  ax-bdsep 14639  ax-infvn 14696
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-suc 4372  df-iom 4591  df-bdc 14596  df-bj-ind 14682
This theorem is referenced by:  bj-findisg  14735  bj-findes  14736
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