Theorem dral1 1741

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.)

Hypothesis

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

Assertion

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

Proof of Theorem dral1

Step	Hyp	Ref	Expression
1		hbae 1729	. . . 4 |- ( A. x x = y -> A. x A. x x = y )
2		dral1.1	. . . . 5 |- ( A. x x = y -> ( ph <-> ps ) )
3	2	biimpd 144	. . . 4 |- ( A. x x = y -> ( ph -> ps ) )
4	1, 3	alimdh 1478	. . 3 |- ( A. x x = y -> ( A. x ph -> A. x ps ) )
5		ax10o 1726	. . 3 |- ( A. x x = y -> ( A. x ps -> A. y ps ) )
6	4, 5	syld 45	. 2 |- ( A. x x = y -> ( A. x ph -> A. y ps ) )
7		hbae 1729	. . . 4 |- ( A. x x = y -> A. y A. x x = y )
8	2	biimprd 158	. . . 4 |- ( A. x x = y -> ( ps -> ph ) )
9	7, 8	alimdh 1478	. . 3 |- ( A. x x = y -> ( A. y ps -> A. y ph ) )
10		ax10o 1726	. . . 4 |- ( A. y y = x -> ( A. y ph -> A. x ph ) )
11	10	alequcoms 1527	. . 3 |- ( A. x x = y -> ( A. y ph -> A. x ph ) )
12	9, 11	syld 45	. 2 |- ( A. x x = y -> ( A. y ps -> A. x ph ) )
13	6, 12	impbid 129	1 |- ( A. x x = y -> ( A. x ph <-> A. y ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  drnf1  1744  equveli  1770  a16g  1875