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Theorem findes 4269
 Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
Hypotheses
Ref Expression
findes.1 [∅ / x]φ
findes.2 (x 𝜔 → (φ[suc x / x]φ))
Assertion
Ref Expression
findes (x 𝜔 → φ)

Proof of Theorem findes
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2761 . 2 (z = ∅ → ([z / x]φ[∅ / x]φ))
2 sbequ 1718 . 2 (z = y → ([z / x]φ ↔ [y / x]φ))
3 dfsbcq2 2761 . 2 (z = suc y → ([z / x]φ[suc y / x]φ))
4 sbequ12r 1652 . 2 (z = x → ([z / x]φφ))
5 findes.1 . 2 [∅ / x]φ
6 nfv 1418 . . . 4 x y 𝜔
7 nfs1v 1812 . . . . 5 x[y / x]φ
8 nfsbc1v 2776 . . . . 5 x[suc y / x]φ
97, 8nfim 1461 . . . 4 x([y / x]φ[suc y / x]φ)
106, 9nfim 1461 . . 3 x(y 𝜔 → ([y / x]φ[suc y / x]φ))
11 eleq1 2097 . . . 4 (x = y → (x 𝜔 ↔ y 𝜔))
12 sbequ12 1651 . . . . 5 (x = y → (φ ↔ [y / x]φ))
13 suceq 4105 . . . . . 6 (x = y → suc x = suc y)
14 dfsbcq 2760 . . . . . 6 (suc x = suc y → ([suc x / x]φ[suc y / x]φ))
1513, 14syl 14 . . . . 5 (x = y → ([suc x / x]φ[suc y / x]φ))
1612, 15imbi12d 223 . . . 4 (x = y → ((φ[suc x / x]φ) ↔ ([y / x]φ[suc y / x]φ)))
1711, 16imbi12d 223 . . 3 (x = y → ((x 𝜔 → (φ[suc x / x]φ)) ↔ (y 𝜔 → ([y / x]φ[suc y / x]φ))))
18 findes.2 . . 3 (x 𝜔 → (φ[suc x / x]φ))
1910, 17, 18chvar 1637 . 2 (y 𝜔 → ([y / x]φ[suc y / x]φ))
201, 2, 3, 4, 5, 19finds 4266 1 (x 𝜔 → φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  [wsb 1642  [wsbc 2758  ∅c0 3218  suc csuc 4068  𝜔com 4256 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257 This theorem is referenced by: (None)
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