Theorem dral1ALT 2175

Description: Alternate proof of dral1 2174, shorter but requiring ax-11 1937. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
dral1.1	|- ( A. x x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
dral1ALT	|- ( A. x x = y -> ( A. x ph <-> A. y ps ) )

Proof of Theorem dral1ALT

Step	Hyp	Ref	Expression
1		dral1.1	. . 3 |- ( A. x x = y -> ( ph <-> ps ) )
2	1	dral2 2173	. 2 |- ( A. x x = y -> ( A. x ph <-> A. x ps ) )
3		axc11 2163	. . 3 |- ( A. x x = y -> ( A. x ps -> A. y ps ) )
4		axc11r 2040	. . 3 |- ( A. x x = y -> ( A. y ps -> A. x ps ) )
5	3, 4	impbid 195	. 2 |- ( A. x x = y -> ( A. x ps <-> A. y ps ) )
6	2, 5	bitrd 261	1 |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676

This theorem is referenced by: (None)