Theorem 19.12vv 1898

Description: Special case of 19.12 1847 where its converse holds. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
19.12vv	|- (E.xA.y(ph -> ps) <-> A.yE.x(ph -> ps))

Distinct variable groups:   ps,x   ph,y

Allowed substitution hints:   ph(x)   ps(y)

Proof of Theorem 19.12vv

Step	Hyp	Ref	Expression
1		19.21v 1890	. . 3 |- (A.y(ph -> ps) <-> (ph -> A.yps))
2	1	exbii 1582	. 2 |- (E.xA.y(ph -> ps) <-> E.x(ph -> A.yps))
3		nfv 1619	. . . 4 |- F/xps
4	3	nfal 1842	. . 3 |- F/xA.yps
5	4	19.36 1871	. 2 |- (E.x(ph -> A.yps) <-> (A.xph -> A.yps))
6		19.36v 1896	. . . 4 |- (E.x(ph -> ps) <-> (A.xph -> ps))
7	6	albii 1566	. . 3 |- (A.yE.x(ph -> ps) <-> A.y(A.xph -> ps))
8		nfv 1619	. . . . 5 |- F/yph
9	8	nfal 1842	. . . 4 |- F/yA.xph
10	9	19.21 1796	. . 3 |- (A.y(A.xph -> ps) <-> (A.xph -> A.yps))
11	7, 10	bitr2i 241	. 2 |- ((A.xph -> A.yps) <-> A.yE.x(ph -> ps))
12	2, 5, 11	3bitri 262	1 |- (E.xA.y(ph -> ps) <-> A.yE.x(ph -> ps))

Syntax hints:   -> wi 4   <-> wb 176  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  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)