Theorem pm10.541 38566

Description: Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm10.541	|- ( A. x ( ph -> ( ch \/ ps ) ) <-> ( ch \/ A. x ( ph -> ps ) ) )

Distinct variable group:   ch, x

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

Proof of Theorem pm10.541

Step	Hyp	Ref	Expression
1		bi2.04 376	. . . 4 |- ( ( ph -> ( -. ch -> ps ) ) <-> ( -. ch -> ( ph -> ps ) ) )
2	1	albii 1747	. . 3 |- ( A. x ( ph -> ( -. ch -> ps ) ) <-> A. x ( -. ch -> ( ph -> ps ) ) )
3		19.21v 1868	. . 3 |- ( A. x ( -. ch -> ( ph -> ps ) ) <-> ( -. ch -> A. x ( ph -> ps ) ) )
4	2, 3	bitri 264	. 2 |- ( A. x ( ph -> ( -. ch -> ps ) ) <-> ( -. ch -> A. x ( ph -> ps ) ) )
5		df-or 385	. . . 4 |- ( ( ch \/ ps ) <-> ( -. ch -> ps ) )
6	5	imbi2i 326	. . 3 |- ( ( ph -> ( ch \/ ps ) ) <-> ( ph -> ( -. ch -> ps ) ) )
7	6	albii 1747	. 2 |- ( A. x ( ph -> ( ch \/ ps ) ) <-> A. x ( ph -> ( -. ch -> ps ) ) )
8		df-or 385	. 2 |- ( ( ch \/ A. x ( ph -> ps ) ) <-> ( -. ch -> A. x ( ph -> ps ) ) )
9	4, 7, 8	3bitr4i 292	1 |- ( A. x ( ph -> ( ch \/ ps ) ) <-> ( ch \/ A. x ( ph -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   A.wal 1481

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

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

This theorem is referenced by: (None)