Theorem r19.29r 2451

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 2450	. 2 |- ( ( A. x e. A ps /\ E. x e. A ph ) -> E. x e. A ( ps /\ ph ) )
2		ancom 253	. 2 |- ( ( E. x e. A ph /\ A. x e. A ps ) <-> ( A. x e. A ps /\ E. x e. A ph ) )
3		ancom 253	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
4	3	rexbii 2331	. 2 |- ( E. x e. A ( ph /\ ps ) <-> E. x e. A ( ps /\ ph ) )
5	1, 2, 4	3imtr4i 190	1 |- ( ( E. x e. A ph /\ A. x e. A ps ) -> E. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wral 2306   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  r19.29af2  2452