Theorem r19.45mv 3528

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 2645	. 2 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )
2		r19.9rmv 3526	. . 3 |- ( E. x x e. A -> ( ph <-> E. x e. A ph ) )
3	2	orbi1d 792	. 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 199	1 |- ( E. x x e. A -> ( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 709   E.wex 1502   e. wcel 2158   E.wrex 2466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-cleq 2180  df-clel 2183  df-rex 2471

This theorem is referenced by:  ltexprlemloc  7620