Theorem tfis2f 2246

Description: Transfinite Induction Schema with implicit substitution.

Hypotheses

Ref	Expression
tfis2f.1	|- (ps -> A.xps)
tfis2f.2	|- (x = y -> (ph <-> ps))
tfis2f.3	|- (x e. On -> (A.y e. x ps -> ph))

Assertion

Ref	Expression
tfis2f	|- (x e. On -> ph)

Distinct variable group(s):   ph,y   x,y

Proof of Theorem tfis2f

Step	Hyp	Ref	Expression
1		tfis2f.3	. . 3 |- (x e. On -> (A.y e. x ps -> ph))
2		tfis2f.1	. . . . 5 |- (ps -> A.xps)
3		tfis2f.2	. . . . 5 |- (x = y -> (ph <-> ps))
4	2, 3	sbie 904	. . . 4 |- ([y / x]ph <-> ps)
5	4	biral 1223	. . 3 |- (A.y e. x [y / x]ph <-> A.y e. x ps)
6	1, 5	syl5ib 181	. 2 |- (x e. On -> (A.y e. x [y / x]ph -> ph))
7	6	tfis 2245	1 |- (x e. On -> ph)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   = weq 797  [wsb 852   e. wcel 1092  A.wral 1201  Oncon0 2199

This theorem is referenced by:  tfis2 2247  tfr3 2964

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

metamath.org