Theorem dral1ALT 2326

Description: Alternate proof of dral1 2325, shorter but requiring ax-11 2034. (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 2324	. 2 |- ( A. x x = y -> ( A. x ph <-> A. x ps ) )
3		axc11 2314	. . 3 |- ( A. x x = y -> ( A. x ps -> A. y ps ) )
4		axc11r 2187	. . 3 |- ( A. x x = y -> ( A. y ps -> A. x ps ) )
5	3, 4	impbid 202	. 2 |- ( A. x x = y -> ( A. x ps <-> A. y ps ) )
6	2, 5	bitrd 268	1 |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 196   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-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)