Theorem tfis3 4252

Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)

Hypotheses

Ref	Expression
tfis3.1	⊢ (x = y → (φ ↔ ψ))
tfis3.2	⊢ (x = A → (φ ↔ χ))
tfis3.3	⊢ (x ∈ On → (∀y ∈ x ψ → φ))

Assertion

Ref	Expression
tfis3	⊢ (A ∈ On → χ)

Distinct variable groups:   ψ,x   φ,y   χ,x   x,A   x,y

Allowed substitution hints:   φ(x)   ψ(y)   χ(y)   A(y)

Proof of Theorem tfis3

Step	Hyp	Ref	Expression
1		tfis3.2	. 2 ⊢ (x = A → (φ ↔ χ))
2		tfis3.1	. . 3 ⊢ (x = y → (φ ↔ ψ))
3		tfis3.3	. . 3 ⊢ (x ∈ On → (∀y ∈ x ψ → φ))
4	2, 3	tfis2 4251	. 2 ⊢ (x ∈ On → φ)
5	1, 4	vtoclga 2613	1 ⊢ (A ∈ On → χ)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Oncon0 4066

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-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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220

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-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071

This theorem is referenced by:  tfisi  4253  tfrlemi1  5887  rdgon  5913