Theorem bnj157 30929

Description: Well-founded induction restricted to a set ( A e. _V). The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj157.1	|- ( ps <-> A. y e. A ( y R x -> [. y / x ]. ph ) )
bnj157.2	|- A e. _V
bnj157.3	|- R Fr A

Assertion

Ref	Expression
bnj157	|- ( A. x e. A ( ps -> ph ) -> A. x e. A ph )

Distinct variable groups:   x, A, y   x, R, y   ph, y

Allowed substitution hints:   ph( x)   ps( x, y)

Proof of Theorem bnj157

Step	Hyp	Ref	Expression
1		bnj157.3	. 2 |- R Fr A
2		bnj157.2	. . 3 |- A e. _V
3		bnj157.1	. . 3 |- ( ps <-> A. y e. A ( y R x -> [. y / x ]. ph ) )
4	2, 3	bnj110 30928	. 2 |- ( ( R Fr A /\ A. x e. A ( ps -> ph ) ) -> A. x e. A ph )
5	1, 4	mpan 706	1 |- ( A. x e. A ( ps -> ph ) -> A. x e. A ph )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   class class class wbr 4653   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-fr 5073

This theorem is referenced by:  bnj852  30991