Theorem setindtrs 37592

Description: Epsilon induction scheme without Infinity. See comments at setindtr 37591. (Contributed by Stefan O'Rear, 28-Oct-2014.)

Hypotheses

Ref	Expression
setindtrs.a	|- ( A. y e. x ps -> ph )
setindtrs.b	|- ( x = y -> ( ph <-> ps ) )
setindtrs.c	|- ( x = B -> ( ph <-> ch ) )

Assertion

Ref	Expression
setindtrs	|- ( E. z ( Tr z /\ B e. z ) -> ch )

Distinct variable groups:   x, B, z   ph, y   ps, x   ch, x   ph, z   x, y

Allowed substitution hints:   ph( x)   ps( y, z)   ch( y, z)   B( y)

Proof of Theorem setindtrs

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setindtr 37591	. . 3 |- ( A. a ( a C_ { x | ph } -> a e. { x | ph } ) -> ( E. z ( Tr z /\ B e. z ) -> B e. { x | ph } ) )
2		dfss3 3592	. . . 4 |- ( a C_ { x | ph } <-> A. y e. a y e. { x | ph } )
3		nfcv 2764	. . . . . . 7 |- F/_ x a
4		nfsab1 2612	. . . . . . 7 |- F/ x y e. { x | ph }
5	3, 4	nfral 2945	. . . . . 6 |- F/ x A. y e. a y e. { x | ph }
6		nfsab1 2612	. . . . . 6 |- F/ x a e. { x | ph }
7	5, 6	nfim 1825	. . . . 5 |- F/ x ( A. y e. a y e. { x | ph } -> a e. { x | ph } )
8		raleq 3138	. . . . . 6 |- ( x = a -> ( A. y e. x y e. { x | ph } <-> A. y e. a y e. { x | ph } ) )
9		eleq1 2689	. . . . . 6 |- ( x = a -> ( x e. { x | ph } <-> a e. { x | ph } ) )
10	8, 9	imbi12d 334	. . . . 5 |- ( x = a -> ( ( A. y e. x y e. { x | ph } -> x e. { x | ph } ) <-> ( A. y e. a y e. { x | ph } -> a e. { x | ph } ) ) )
11		setindtrs.a	. . . . . 6 |- ( A. y e. x ps -> ph )
12		vex 3203	. . . . . . . 8 |- y e. _V
13		setindtrs.b	. . . . . . . 8 |- ( x = y -> ( ph <-> ps ) )
14	12, 13	elab 3350	. . . . . . 7 |- ( y e. { x | ph } <-> ps )
15	14	ralbii 2980	. . . . . 6 |- ( A. y e. x y e. { x | ph } <-> A. y e. x ps )
16		abid 2610	. . . . . 6 |- ( x e. { x | ph } <-> ph )
17	11, 15, 16	3imtr4i 281	. . . . 5 |- ( A. y e. x y e. { x | ph } -> x e. { x | ph } )
18	7, 10, 17	chvar 2262	. . . 4 |- ( A. y e. a y e. { x | ph } -> a e. { x | ph } )
19	2, 18	sylbi 207	. . 3 |- ( a C_ { x | ph } -> a e. { x | ph } )
20	1, 19	mpg 1724	. 2 |- ( E. z ( Tr z /\ B e. z ) -> B e. { x | ph } )
21		elex 3212	. . . . 5 |- ( B e. z -> B e. _V )
22	21	adantl 482	. . . 4 |- ( ( Tr z /\ B e. z ) -> B e. _V )
23	22	exlimiv 1858	. . 3 |- ( E. z ( Tr z /\ B e. z ) -> B e. _V )
24		setindtrs.c	. . . 4 |- ( x = B -> ( ph <-> ch ) )
25	24	elabg 3351	. . 3 |- ( B e. _V -> ( B e. { x | ph } <-> ch ) )
26	23, 25	syl 17	. 2 |- ( E. z ( Tr z /\ B e. z ) -> ( B e. { x | ph } <-> ch ) )
27	20, 26	mpbid 222	1 |- ( E. z ( Tr z /\ B e. z ) -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   C_ wss 3574   Tr wtr 4752

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  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  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-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-uni 4437  df-tr 4753

This theorem is referenced by:  dford3lem2  37594