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Theorem tfis2f 7206
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
Hypotheses
Ref Expression
tfis2f.1 𝑥𝜓
tfis2f.2 (𝑥 = 𝑦 → (𝜑𝜓))
tfis2f.3 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
Assertion
Ref Expression
tfis2f (𝑥 ∈ On → 𝜑)
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem tfis2f
StepHypRef Expression
1 tfis2f.1 . . . . 5 𝑥𝜓
2 tfis2f.2 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2sbie 2555 . . . 4 ([𝑦 / 𝑥]𝜑𝜓)
43ralbii 3129 . . 3 (∀𝑦𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦𝑥 𝜓)
5 tfis2f.3 . . 3 (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))
64, 5syl5bi 232 . 2 (𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))
76tfis 7205 1 (𝑥 ∈ On → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1856  [wsb 2049  wcel 2145  wral 3061  Oncon0 5865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5868  df-on 5869
This theorem is referenced by:  tfis2  7207  tfr3  7652
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