Theorem r19.29 2755

Description: Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

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

Proof of Theorem r19.29

Step	Hyp	Ref	Expression
1		pm3.2 434	. . . 4 |- (ph -> (ps -> (ph /\ ps)))
2	1	ralimi 2690	. . 3 |- (A.x e. A ph -> A.x e. A (ps -> (ph /\ ps)))
3		rexim 2719	. . 3 |- (A.x e. A (ps -> (ph /\ ps)) -> (E.x e. A ps -> E.x e. A (ph /\ ps)))
4	2, 3	syl 15	. 2 |- (A.x e. A ph -> (E.x e. A ps -> E.x e. A (ph /\ ps)))
5	4	imp 418	1 |- ((A.x e. A ph /\ E.x e. A ps) -> E.x e. A (ph /\ ps))

Syntax hints:   -> wi 4   /\ wa 358  A.wral 2615  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2620  df-rex 2621

This theorem is referenced by:  r19.29r  2756  fun11iun  5306  fmpt  5693