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Theorem findes 4251
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 2742 . 2 (z = ∅ → ([z / x]φ[∅ / x]φ))
2 sbequ 1703 . 2 (z = y → ([z / x]φ ↔ [y / x]φ))
3 dfsbcq2 2742 . 2 (z = suc y → ([z / x]φ[suc y / x]φ))
4 sbequ12r 1637 . 2 (z = x → ([z / x]φφ))
5 findes.1 . 2 [∅ / x]φ
6 nfv 1402 . . . 4 x y 𝜔
7 nfs1v 1797 . . . . 5 x[y / x]φ
8 nfsbc1v 2757 . . . . 5 x[suc y / x]φ
97, 8nfim 1446 . . . 4 x([y / x]φ[suc y / x]φ)
106, 9nfim 1446 . . 3 x(y 𝜔 → ([y / x]φ[suc y / x]φ))
11 eleq1 2082 . . . 4 (x = y → (x 𝜔 ↔ y 𝜔))
12 sbequ12 1636 . . . . 5 (x = y → (φ ↔ [y / x]φ))
13 suceq 4086 . . . . . 6 (x = y → suc x = suc y)
14 dfsbcq 2741 . . . . . 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 1622 . 2 (y 𝜔 → ([y / x]φ[suc y / x]φ))
201, 2, 3, 4, 5, 19finds 4248 1 (x 𝜔 → φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  [wsb 1627  [wsbc 2739  c0 3199  suc csuc 4049  𝜔com 4238
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-nul 3855  ax-pow 3899  ax-pr 3916  ax-un 4118  ax-iinf 4236
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-dif 2895  df-un 2897  df-in 2899  df-ss 2906  df-nul 3200  df-pw 3334  df-sn 3354  df-pr 3355  df-uni 3553  df-int 3588  df-suc 4055  df-iom 4239
This theorem is referenced by: (None)
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