Theorem r19.29r 2632

Description: Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)

Assertion

Ref	Expression
r19.29r	|- ( ( E. x e. A ph /\ A. x e. A ps ) -> E. x e. A ( ph /\ ps ) )

Proof of Theorem r19.29r

Step	Hyp	Ref	Expression
1		r19.29 2631	. 2 |- ( ( A. x e. A ps /\ E. x e. A ph ) -> E. x e. A ( ps /\ ph ) )
2		ancom 266	. 2 |- ( ( E. x e. A ph /\ A. x e. A ps ) <-> ( A. x e. A ps /\ E. x e. A ph ) )
3		ancom 266	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
4	3	rexbii 2501	. 2 |- ( E. x e. A ( ph /\ ps ) <-> E. x e. A ( ps /\ ph ) )
5	1, 2, 4	3imtr4i 201	1 |- ( ( E. x e. A ph /\ A. x e. A ps ) -> E. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2472   E.wrex 2473

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-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-ral 2477  df-rex 2478

This theorem is referenced by:  r19.29af2  2634  lmss  14414  metcnp3  14679