Theorem 19.43 1605

Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)

Assertion

Ref	Expression
19.43	|- (E.x(ph \/ ps) <-> (E.xph \/ E.xps))

Proof of Theorem 19.43

Step	Hyp	Ref	Expression
1		df-or 359	. . . 4 |- ((ph \/ ps) <-> (-. ph -> ps))
2	1	exbii 1582	. . 3 |- (E.x(ph \/ ps) <-> E.x(-. ph -> ps))
3		19.35 1600	. . 3 |- (E.x(-. ph -> ps) <-> (A.x -. ph -> E.xps))
4		alnex 1543	. . . 4 |- (A.x -. ph <-> -. E.xph)
5	4	imbi1i 315	. . 3 |- ((A.x -. ph -> E.xps) <-> (-. E.xph -> E.xps))
6	2, 3, 5	3bitri 262	. 2 |- (E.x(ph \/ ps) <-> (-. E.xph -> E.xps))
7		df-or 359	. 2 |- ((E.xph \/ E.xps) <-> (-. E.xph -> E.xps))
8	6, 7	bitr4i 243	1 |- (E.x(ph \/ ps) <-> (E.xph \/ E.xps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357  A.wal 1540  E.wex 1541

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-or 359  df-an 360  df-ex 1542

This theorem is referenced by:  19.34  1663  19.44  1877  19.45  1878  rexun  3444  unipr  3906  uniun  3911  opeq  4620  unopab  4639  dmun  4913  coundi  5083  coundir  5084