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Theorem finds2 4266
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
Hypotheses
Ref Expression
finds2.1 (x = ∅ → (φψ))
finds2.2 (x = y → (φχ))
finds2.3 (x = suc y → (φθ))
finds2.4 (τψ)
finds2.5 (y 𝜔 → (τ → (χθ)))
Assertion
Ref Expression
finds2 (x 𝜔 → (τφ))
Distinct variable groups:   x,y,τ   ψ,x   χ,x   θ,x   φ,y
Allowed substitution hints:   φ(x)   ψ(y)   χ(y)   θ(y)

Proof of Theorem finds2
StepHypRef Expression
1 finds2.4 . . . . 5 (τψ)
2 0ex 3874 . . . . . 6 V
3 finds2.1 . . . . . . 7 (x = ∅ → (φψ))
43imbi2d 219 . . . . . 6 (x = ∅ → ((τφ) ↔ (τψ)))
52, 4elab 2681 . . . . 5 (∅ {x ∣ (τφ)} ↔ (τψ))
61, 5mpbir 134 . . . 4 {x ∣ (τφ)}
7 finds2.5 . . . . . . 7 (y 𝜔 → (τ → (χθ)))
87a2d 23 . . . . . 6 (y 𝜔 → ((τχ) → (τθ)))
9 vex 2554 . . . . . . 7 y V
10 finds2.2 . . . . . . . 8 (x = y → (φχ))
1110imbi2d 219 . . . . . . 7 (x = y → ((τφ) ↔ (τχ)))
129, 11elab 2681 . . . . . 6 (y {x ∣ (τφ)} ↔ (τχ))
139sucex 4190 . . . . . . 7 suc y V
14 finds2.3 . . . . . . . 8 (x = suc y → (φθ))
1514imbi2d 219 . . . . . . 7 (x = suc y → ((τφ) ↔ (τθ)))
1613, 15elab 2681 . . . . . 6 (suc y {x ∣ (τφ)} ↔ (τθ))
178, 12, 163imtr4g 194 . . . . 5 (y 𝜔 → (y {x ∣ (τφ)} → suc y {x ∣ (τφ)}))
1817rgen 2368 . . . 4 y 𝜔 (y {x ∣ (τφ)} → suc y {x ∣ (τφ)})
19 peano5 4263 . . . 4 ((∅ {x ∣ (τφ)} y 𝜔 (y {x ∣ (τφ)} → suc y {x ∣ (τφ)})) → 𝜔 ⊆ {x ∣ (τφ)})
206, 18, 19mp2an 402 . . 3 𝜔 ⊆ {x ∣ (τφ)}
2120sseli 2935 . 2 (x 𝜔 → x {x ∣ (τφ)})
22 abid 2025 . 2 (x {x ∣ (τφ)} ↔ (τφ))
2321, 22sylib 127 1 (x 𝜔 → (τφ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wss 2911  c0 3218  suc csuc 4067  𝜔com 4255
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-bnd 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 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-iinf 4253
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-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-uni 3571  df-int 3606  df-suc 4073  df-iom 4256
This theorem is referenced by:  finds1  4267  frecrdg  5925  nnacl  5991  nnmcl  5992  nnacom  5995  nnaass  5996  nndi  5997  nnmass  5998  nnmsucr  5999  nnmcom  6000  nnsucsssuc  6003  nntri3or  6004  nnaordi  6010  nnaword  6013  nnmordi  6018  nnaordex  6029  prarloclem3  6472  frec2uzzd  8813  frec2uzuzd  8815
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