Theorem dral1ALT 2162

Description: Alternate proof of dral1 2161, shorter but requiring ax-11 1922. (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 2160	. 2 |- ( A. x x = y -> ( A. x ph <-> A. x ps ) )
3		axc11 2150	. . 3 |- ( A. x x = y -> ( A. x ps -> A. y ps ) )
4		axc112 2022	. . 3 |- ( A. x x = y -> ( A. y ps -> A. x ps ) )
5	3, 4	impbid 194	. 2 |- ( A. x x = y -> ( A. x ps <-> A. y ps ) )
6	2, 5	bitrd 257	1 |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 188   A.wal 1444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1666  df-nf 1670

This theorem is referenced by: (None)