Theorem axi5r 2594

Description: Converse of ax-c4 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)

Assertion

Ref	Expression
axi5r	|- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> ps ) )

Proof of Theorem axi5r

Step	Hyp	Ref	Expression
1		hba1 2151	. . 3 |- ( A. x ph -> A. x A. x ph )
2		hba1 2151	. . 3 |- ( A. x ps -> A. x A. x ps )
3	1, 2	hbim 2127	. 2 |- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> A. x ps ) )
4		sp 2053	. . 3 |- ( A. x ps -> ps )
5	4	imim2i 16	. 2 |- ( ( A. x ph -> A. x ps ) -> ( A. x ph -> ps ) )
6	3, 5	alrimih 1751	1 |- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   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  ax-6 1888  ax-7 1935  ax-10 2019  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)