Theorem r19.45av 2769

Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)

Assertion

Ref	Expression
r19.45av	|- (E.x e. A (ph \/ ps) -> (ph \/ E.x e. A ps))

Distinct variable group:   ph,x

Allowed substitution hints:   ps(x)   A(x)

Proof of Theorem r19.45av

Step	Hyp	Ref	Expression
1		r19.43 2767	. 2 |- (E.x e. A (ph \/ ps) <-> (E.x e. A ph \/ E.x e. A ps))
2		idd 21	. . . 4 |- (x e. A -> (ph -> ph))
3	2	rexlimiv 2733	. . 3 |- (E.x e. A ph -> ph)
4	3	orim1i 503	. 2 |- ((E.x e. A ph \/ E.x e. A ps) -> (ph \/ E.x e. A ps))
5	1, 4	sylbi 187	1 |- (E.x e. A (ph \/ ps) -> (ph \/ E.x e. A ps))

Syntax hints:   -> wi 4   \/ wo 357   e. wcel 1710  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  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by: (None)