Theorem r19.44mv 3457

Description: Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
r19.44mv	|- ( E. y y e. A -> ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ ps ) ) )

Distinct variable groups:   x, A   y, A   ps, x

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

Proof of Theorem r19.44mv

Step	Hyp	Ref	Expression
1		r19.9rmv 3454	. . 3 |- ( E. y y e. A -> ( ps <-> E. x e. A ps ) )
2	1	orbi2d 779	. 2 |- ( E. y y e. A -> ( ( E. x e. A ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) ) )
3		r19.43 2589	. 2 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )
4	2, 3	syl6rbbr 198	1 |- ( E. y y e. A -> ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 697   E.wex 1468   e. wcel 1480   E.wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135  df-rex 2422

This theorem is referenced by:  frecabcl  6296