Theorem r19.36av 2760

Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. The other direction doesn't hold when A is empty. (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
r19.36av	|- (E.x e. A (ph -> ps) -> (A.x e. A ph -> ps))

Distinct variable group:   ps,x

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

Proof of Theorem r19.36av

Step	Hyp	Ref	Expression
1		r19.35 2759	. 2 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))
2		idd 21	. . . 4 |- (x e. A -> (ps -> ps))
3	2	rexlimiv 2733	. . 3 |- (E.x e. A ps -> ps)
4	3	imim2i 13	. 2 |- ((A.x e. A ph -> E.x e. A ps) -> (A.x e. A ph -> ps))
5	1, 4	sylbi 187	1 |- (E.x e. A (ph -> ps) -> (A.x e. A ph -> ps))

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

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

This theorem is referenced by:  iinss  4018