Theorem drex1 1967

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.)

Hypothesis

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

Assertion

Ref	Expression
drex1	|- (A.x x = y -> (E.xph <-> E.yps))

Proof of Theorem drex1

Step	Hyp	Ref	Expression
1		dral1.1	. . . . 5 |- (A.x x = y -> (ph <-> ps))
2	1	notbid 285	. . . 4 |- (A.x x = y -> (-. ph <-> -. ps))
3	2	dral1 1965	. . 3 |- (A.x x = y -> (A.x -. ph <-> A.y -. ps))
4	3	notbid 285	. 2 |- (A.x x = y -> (-. A.x -. ph <-> -. A.y -. ps))
5		df-ex 1542	. 2 |- (E.xph <-> -. A.x -. ph)
6		df-ex 1542	. 2 |- (E.yps <-> -. A.y -. ps)
7	4, 5, 6	3bitr4g 279	1 |- (A.x x = y -> (E.xph <-> E.yps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  exdistrf  1971  drsb1  2022  eujustALT  2207  copsexg  4608  dfid3  4769