Theorem dral1-o 34189

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2325 using ax-c11 34172. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

Proof of Theorem dral1-o

Step	Hyp	Ref	Expression
1		hbae-o 34188	. . . 4 |- ( A. x x = y -> A. x A. x x = y )
2		dral1-o.1	. . . . 5 |- ( A. x x = y -> ( ph <-> ps ) )
3	2	biimpd 219	. . . 4 |- ( A. x x = y -> ( ph -> ps ) )
4	1, 3	alimdh 1745	. . 3 |- ( A. x x = y -> ( A. x ph -> A. x ps ) )
5		ax-c11 34172	. . 3 |- ( A. x x = y -> ( A. x ps -> A. y ps ) )
6	4, 5	syld 47	. 2 |- ( A. x x = y -> ( A. x ph -> A. y ps ) )
7		hbae-o 34188	. . . 4 |- ( A. x x = y -> A. y A. x x = y )
8	2	biimprd 238	. . . 4 |- ( A. x x = y -> ( ps -> ph ) )
9	7, 8	alimdh 1745	. . 3 |- ( A. x x = y -> ( A. y ps -> A. y ph ) )
10		ax-c11 34172	. . . 4 |- ( A. y y = x -> ( A. y ph -> A. x ph ) )
11	10	aecoms-o 34187	. . 3 |- ( A. x x = y -> ( A. y ph -> A. x ph ) )
12	9, 11	syld 47	. 2 |- ( A. x x = y -> ( A. y ps -> A. x ph ) )
13	6, 12	impbid 202	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-11 2034  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  ax12fromc15  34190  axc16g-o  34219  ax12indalem  34230  ax12inda2ALT  34231