Theorem dral1 1137

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).

Hypothesis

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

Assertion

Ref	Expression
dral1	|- (A.x x = y -> (A.xph <-> A.yps))

Proof of Theorem dral1

Step	Hyp	Ref	Expression
1		dral1.1	. . . . . 6 |- (A.x x = y -> (ph <-> ps))
2	1	biimpd 153	. . . . 5 |- (A.x x = y -> (ph -> ps))
3	2	19.20ii 971	. . . 4 |- (A.xA.x x = y -> (A.xph -> A.xps))
4	3	hbaes 1129	. . 3 |- (A.x x = y -> (A.xph -> A.xps))
5		ax-10 1103	. . 3 |- (A.x x = y -> (A.xps -> A.yps))
6	4, 5	syld 27	. 2 |- (A.x x = y -> (A.xph -> A.yps))
7	1	biimprd 154	. . . . 5 |- (A.x x = y -> (ps -> ph))
8	7	19.20ii 971	. . . 4 |- (A.yA.x x = y -> (A.yps -> A.yph))
9	8	hbaes 1129	. . 3 |- (A.x x = y -> (A.yps -> A.yph))
10		ax-10 1103	. . . 4 |- (A.y y = x -> (A.yph -> A.xph))
11	10	alequcoms 1126	. . 3 |- (A.x x = y -> (A.yph -> A.xph))
12	9, 11	syld 27	. 2 |- (A.x x = y -> (A.yps -> A.xph))
13	6, 12	impbid 514	1 |- (A.x x = y -> (A.xph <-> A.yps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 950   = wceq 1099

This theorem is referenced by:  drex1 1139  ax11 1203  hbsb4 1232  sb9i 1247  a16g 1258  ax11indalem 1345  ax11inda2ALT 1346  ralcom2 1752  axpownd 4876

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957

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