Theorem pm11.53 2179

Description: Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1906 for a version requiring fewer axioms. (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 1868	. . 3 |- ( A. y ( ph -> ps ) <-> ( ph -> A. y ps ) )
2	1	albii 1747	. 2 |- ( A. x A. y ( ph -> ps ) <-> A. x ( ph -> A. y ps ) )
3		nfv 1843	. . . 4 |- F/ x ps
4	3	nfal 2153	. . 3 |- F/ x A. y ps
5	4	19.23 2080	. 2 |- ( A. x ( ph -> A. y ps ) <-> ( E. x ph -> A. y ps ) )
6	2, 5	bitri 264	1 |- ( A. x A. y ( ph -> ps ) <-> ( E. x ph -> A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)