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Theorem bdsetindis 10907
Description: Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdsetindis.bd BOUNDED 𝜑
bdsetindis.nf0 𝑥𝜓
bdsetindis.nf1 𝑥𝜒
bdsetindis.nf2 𝑦𝜑
bdsetindis.nf3 𝑦𝜓
bdsetindis.1 (𝑥 = 𝑧 → (𝜑𝜓))
bdsetindis.2 (𝑥 = 𝑦 → (𝜒𝜑))
Assertion
Ref Expression
bdsetindis (∀𝑦(∀𝑧𝑦 𝜓𝜒) → ∀𝑥𝜑)
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem bdsetindis
StepHypRef Expression
1 nfcv 2220 . . . . 5 𝑥𝑦
2 bdsetindis.nf0 . . . . 5 𝑥𝜓
31, 2nfralxy 2403 . . . 4 𝑥𝑧𝑦 𝜓
4 bdsetindis.nf1 . . . 4 𝑥𝜒
53, 4nfim 1505 . . 3 𝑥(∀𝑧𝑦 𝜓𝜒)
6 nfcv 2220 . . . . 5 𝑦𝑥
7 bdsetindis.nf3 . . . . 5 𝑦𝜓
86, 7nfralxy 2403 . . . 4 𝑦𝑧𝑥 𝜓
9 bdsetindis.nf2 . . . 4 𝑦𝜑
108, 9nfim 1505 . . 3 𝑦(∀𝑧𝑥 𝜓𝜑)
11 raleq 2550 . . . . 5 (𝑦 = 𝑥 → (∀𝑧𝑦 𝜓 ↔ ∀𝑧𝑥 𝜓))
1211biimprd 156 . . . 4 (𝑦 = 𝑥 → (∀𝑧𝑥 𝜓 → ∀𝑧𝑦 𝜓))
13 bdsetindis.2 . . . . 5 (𝑥 = 𝑦 → (𝜒𝜑))
1413equcoms 1635 . . . 4 (𝑦 = 𝑥 → (𝜒𝜑))
1512, 14imim12d 73 . . 3 (𝑦 = 𝑥 → ((∀𝑧𝑦 𝜓𝜒) → (∀𝑧𝑥 𝜓𝜑)))
165, 10, 15cbv3 1671 . 2 (∀𝑦(∀𝑧𝑦 𝜓𝜒) → ∀𝑥(∀𝑧𝑥 𝜓𝜑))
17 bdsetindis.1 . . . . . 6 (𝑥 = 𝑧 → (𝜑𝜓))
182, 17bj-sbime 10720 . . . . 5 ([𝑧 / 𝑥]𝜑𝜓)
1918ralimi 2427 . . . 4 (∀𝑧𝑥 [𝑧 / 𝑥]𝜑 → ∀𝑧𝑥 𝜓)
2019imim1i 59 . . 3 ((∀𝑧𝑥 𝜓𝜑) → (∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑))
2120alimi 1385 . 2 (∀𝑥(∀𝑧𝑥 𝜓𝜑) → ∀𝑥(∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑))
22 bdsetindis.bd . . 3 BOUNDED 𝜑
2322ax-bdsetind 10906 . 2 (∀𝑥(∀𝑧𝑥 [𝑧 / 𝑥]𝜑𝜑) → ∀𝑥𝜑)
2416, 21, 233syl 17 1 (∀𝑦(∀𝑧𝑦 𝜓𝜒) → ∀𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390  [wsb 1686  wral 2349  BOUNDED wbd 10746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-bdsetind 10906
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354
This theorem is referenced by:  bj-inf2vnlem3  10910
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