Theorem r19.45mv 3335

Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
r19.45mv	|- ( E. x x e. A -> ( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) )

Distinct variable groups:   x, A   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem r19.45mv

Step	Hyp	Ref	Expression
1		r19.9rmv 3333	. . 3 |- ( E. x x e. A -> ( ph <-> E. x e. A ph ) )
2	1	orbi1d 737	. 2 |- ( E. x x e. A -> ( ( ph \/ E. x e. A ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) ) )
3		r19.43 2512	. 2 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )
4	2, 3	syl6rbbr 197	1 |- ( E. x x e. A -> ( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) )

Syntax hints:   -> wi 4   <-> wb 103   \/ wo 661   E.wex 1421   e. wcel 1433   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-rex 2354

This theorem is referenced by:  ltexprlemloc  6797