Theorem r19.27av 2224

Description: Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (The proof was shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.27av	|- ((A.x e. A ph /\ ps) -> A.x e. A (ph /\ ps))

Distinct variable group:   ps,x

Proof of Theorem r19.27av

Step	Hyp	Ref	Expression
1		ax-1 4	. . . 4 |- (ps -> (x e. A -> ps))
2	1	r19.21aiv 2175	. . 3 |- (ps -> A.x e. A ps)
3	2	anim2i 362	. 2 |- ((A.x e. A ph /\ ps) -> (A.x e. A ph /\ A.x e. A ps))
4		r19.26 2219	. 2 |- (A.x e. A (ph /\ ps) <-> (A.x e. A ph /\ A.x e. A ps))
5	3, 4	sylibr 217	1 |- ((A.x e. A ph /\ ps) -> A.x e. A (ph /\ ps))

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  A.wral 2105

This theorem is referenced by:  r19.28av 2226  spanuni 11100  intartar 15255  tartarmap 15265

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-an 242  df-ral 2109

Copyright terms: Public domain