Theorem r19.45mv 3508

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		r19.43 2628	. 2 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )
2		r19.9rmv 3506	. . 3 |- ( E. x x e. A -> ( ph <-> E. x e. A ph ) )
3	2	orbi1d 786	. 2 |- ( E. x x e. A -> ( ( ph \/ E. x e. A ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) ) )
4	1, 3	bitr4id 198	1 |- ( E. x x e. A -> ( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 703   E.wex 1485   e. wcel 2141   E.wrex 2449

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-rex 2454

This theorem is referenced by:  ltexprlemloc  7569