Theorem bnj1148 31064

Description: Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1148	|- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) e. _V )

Proof of Theorem bnj1148

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 3215	. . . . 5 |- ( X e. A -> E. x x = X )
2	1	adantl 482	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> E. x x = X )
3		bnj93 30933	. . . . 5 |- ( ( R FrSe A /\ x e. A ) -> pred ( x , A , R ) e. _V )
4		eleq1 2689	. . . . . . 7 |- ( x = X -> ( x e. A <-> X e. A ) )
5	4	anbi2d 740	. . . . . 6 |- ( x = X -> ( ( R FrSe A /\ x e. A ) <-> ( R FrSe A /\ X e. A ) ) )
6		bnj602 30985	. . . . . . 7 |- ( x = X -> pred ( x , A , R ) = pred ( X , A , R ) )
7	6	eleq1d 2686	. . . . . 6 |- ( x = X -> ( pred ( x , A , R ) e. _V <-> pred ( X , A , R ) e. _V ) )
8	5, 7	imbi12d 334	. . . . 5 |- ( x = X -> ( ( ( R FrSe A /\ x e. A ) -> pred ( x , A , R ) e. _V ) <-> ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) e. _V ) ) )
9	3, 8	mpbii 223	. . . 4 |- ( x = X -> ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) e. _V ) )
10	2, 9	bnj593 30815	. . 3 |- ( ( R FrSe A /\ X e. A ) -> E. x ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) e. _V ) )
11	10	bnj937 30842	. 2 |- ( ( R FrSe A /\ X e. A ) -> ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) e. _V ) )
12	11	pm2.43i 52	1 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) e. _V )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   predc-bnj14 30754   FrSe w-bnj15 30758

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-ral 2917  df-rab 2921  df-v 3202  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-bnj14 30755  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj1136  31065  bnj1413  31103