Theorem 19.35 1600

Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)

Assertion

Ref	Expression
19.35	|- (E.x(ph -> ps) <-> (A.xph -> E.xps))

Proof of Theorem 19.35

Step	Hyp	Ref	Expression
1		19.26 1593	. . . 4 |- (A.x(ph /\ -. ps) <-> (A.xph /\ A.x -. ps))
2		annim 414	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	albii 1566	. . . 4 |- (A.x(ph /\ -. ps) <-> A.x -. (ph -> ps))
4		alnex 1543	. . . . 5 |- (A.x -. ps <-> -. E.xps)
5	4	anbi2i 675	. . . 4 |- ((A.xph /\ A.x -. ps) <-> (A.xph /\ -. E.xps))
6	1, 3, 5	3bitr3i 266	. . 3 |- (A.x -. (ph -> ps) <-> (A.xph /\ -. E.xps))
7		alnex 1543	. . 3 |- (A.x -. (ph -> ps) <-> -. E.x(ph -> ps))
8		annim 414	. . 3 |- ((A.xph /\ -. E.xps) <-> -. (A.xph -> E.xps))
9	6, 7, 8	3bitr3i 266	. 2 |- (-. E.x(ph -> ps) <-> -. (A.xph -> E.xps))
10	9	con4bii 288	1 |- (E.x(ph -> ps) <-> (A.xph -> E.xps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  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-an 360  df-ex 1542

This theorem is referenced by:  19.35i  1601  19.35ri  1602  19.25  1603  19.43  1605  speimfw  1645  19.39  1661  19.24  1662  19.36  1871  19.37  1873  sbequi  2059