Theorem predbrg 5700

Description: Closed form of elpredim 5692. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)

Assertion

Ref	Expression
predbrg	|- ( ( X e. V /\ Y e. Pred ( R , A , X ) ) -> Y R X )

Proof of Theorem predbrg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		predeq3 5684	. . . . 5 |- ( x = X -> Pred ( R , A , x ) = Pred ( R , A , X ) )
2	1	eleq2d 2687	. . . 4 |- ( x = X -> ( Y e. Pred ( R , A , x ) <-> Y e. Pred ( R , A , X ) ) )
3		breq2 4657	. . . 4 |- ( x = X -> ( Y R x <-> Y R X ) )
4	2, 3	imbi12d 334	. . 3 |- ( x = X -> ( ( Y e. Pred ( R , A , x ) -> Y R x ) <-> ( Y e. Pred ( R , A , X ) -> Y R X ) ) )
5		vex 3203	. . . 4 |- x e. _V
6	5	elpredim 5692	. . 3 |- ( Y e. Pred ( R , A , x ) -> Y R x )
7	4, 6	vtoclg 3266	. 2 |- ( X e. V -> ( Y e. Pred ( R , A , X ) -> Y R X ) )
8	7	imp 445	1 |- ( ( X e. V /\ Y e. Pred ( R , A , X ) ) -> Y R X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   Predcpred 5679

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-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by: (None)