Theorem bnj93 30933

Description: Technical lemma for bnj97 30936. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj93	|- ( ( R FrSe A /\ x e. A ) -> pred ( x , A , R ) e. _V )

Distinct variable groups:   x, A   x, R

Proof of Theorem bnj93

Step	Hyp	Ref	Expression
1		df-bnj15 30759	. . . 4 |- ( R FrSe A <-> ( R Fr A /\ R Se A ) )
2	1	simprbi 480	. . 3 |- ( R FrSe A -> R Se A )
3		df-bnj13 30757	. . 3 |- ( R Se A <-> A. x e. A pred ( x , A , R ) e. _V )
4	2, 3	sylib 208	. 2 |- ( R FrSe A -> A. x e. A pred ( x , A , R ) e. _V )
5	4	r19.21bi 2932	1 |- ( ( R FrSe A /\ x e. A ) -> pred ( x , A , R ) e. _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   _Vcvv 3200   Fr wfr 5070   predc-bnj14 30754   Se w-bnj13 30756   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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj96  30935  bnj97  30936  bnj149  30945  bnj150  30946  bnj518  30956  bnj1148  31064