Theorem pm11.53 1907

Description: Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.53	|- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )

Distinct variable groups:   ph, y   ps, x

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

Proof of Theorem pm11.53

Step	Hyp	Ref	Expression
1		19.21v 1884	. . 3 |- ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
2	1	albii 1481	. 2 |- ( A. x A. y ( ph -> ps ) <-> A. x ( ph -> A. y ps ) )
3		ax-17 1537	. . . 4 |- ( ps -> A. x ps )
4	3	hbal 1488	. . 3 |- ( A. y ps -> A. x A. y ps )
5	4	19.23h 1509	. 2 |- ( A. x ( ph -> A. y ps ) <-> ( E. x ph -> A. y ps ) )
6	2, 5	bitri 184	1 |- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   E.wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)