Theorem drex1 1809

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, 27-Feb-2005.) (Revised by NM, 3-Feb-2015.)

Hypothesis

Ref	Expression
drex1.1	|- ( A. x x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
drex1	|- ( A. x x = y -> ( E. x ph <-> E. y ps ) )

Proof of Theorem drex1

Step	Hyp	Ref	Expression
1		hbae 1729	. . . 4 |- ( A. x x = y -> A. x A. x x = y )
2		drex1.1	. . . . 5 |- ( A. x x = y -> ( ph <-> ps ) )
3		ax-4 1521	. . . . . 6 |- ( A. x x = y -> x = y )
4	3	biantrurd 305	. . . . 5 |- ( A. x x = y -> ( ps <-> ( x = y /\ ps ) ) )
5	2, 4	bitr2d 189	. . . 4 |- ( A. x x = y -> ( ( x = y /\ ps ) <-> ph ) )
6	1, 5	exbidh 1625	. . 3 |- ( A. x x = y -> ( E. x ( x = y /\ ps ) <-> E. x ph ) )
7		ax11e 1807	. . . 4 |- ( x = y -> ( E. x ( x = y /\ ps ) -> E. y ps ) )
8	7	sps 1548	. . 3 |- ( A. x x = y -> ( E. x ( x = y /\ ps ) -> E. y ps ) )
9	6, 8	sylbird 170	. 2 |- ( A. x x = y -> ( E. x ph -> E. y ps ) )
10		hbae 1729	. . . 4 |- ( A. x x = y -> A. y A. x x = y )
11		equcomi 1715	. . . . . . 7 |- ( x = y -> y = x )
12	11	sps 1548	. . . . . 6 |- ( A. x x = y -> y = x )
13	12	biantrurd 305	. . . . 5 |- ( A. x x = y -> ( ph <-> ( y = x /\ ph ) ) )
14	13, 2	bitr3d 190	. . . 4 |- ( A. x x = y -> ( ( y = x /\ ph ) <-> ps ) )
15	10, 14	exbidh 1625	. . 3 |- ( A. x x = y -> ( E. y ( y = x /\ ph ) <-> E. y ps ) )
16		ax11e 1807	. . . . 5 |- ( y = x -> ( E. y ( y = x /\ ph ) -> E. x ph ) )
17	16	sps 1548	. . . 4 |- ( A. y y = x -> ( E. y ( y = x /\ ph ) -> E. x ph ) )
18	17	alequcoms 1527	. . 3 |- ( A. x x = y -> ( E. y ( y = x /\ ph ) -> E. x ph ) )
19	15, 18	sylbird 170	. 2 |- ( A. x x = y -> ( E. y ps -> E. x ph ) )
20	9, 19	impbid 129	1 |- ( A. x x = y -> ( E. x ph <-> E. y ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   E.wex 1503

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-ie1 1504  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:  drsb1  1810  exdistrfor  1811  copsexg  4273