Theorem pm11.53 1823

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 1801	. . 3 |- ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
2	1	albii 1404	. 2 |- ( A. x A. y ( ph -> ps ) <-> A. x ( ph -> A. y ps ) )
3		ax-17 1464	. . . 4 |- ( ps -> A. x ps )
4	3	hbal 1411	. . 3 |- ( A. y ps -> A. x A. y ps )
5	4	19.23h 1432	. 2 |- ( A. x ( ph -> A. y ps ) <-> ( E. x ph -> A. y ps ) )
6	2, 5	bitri 182	1 |- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1287   E.wex 1426

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)