Theorem dral1 1154

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		hbae 1145	. . . 4 |- (A.x x = y -> A.xA.x x = y)
2		dral1.1	. . . . 5 |- (A.x x = y -> (ph <-> ps))
3	2	biimpd 153	. . . 4 |- (A.x x = y -> (ph -> ps))
4	1, 3	19.20d 996	. . 3 |- (A.x x = y -> (A.xph -> A.xps))
5		ax-10o 1140	. . 3 |- (A.x x = y -> (A.xps -> A.yps))
6	4, 5	syld 27	. 2 |- (A.x x = y -> (A.xph -> A.yps))
7		hbae 1145	. . . 4 |- (A.x x = y -> A.yA.x x = y)
8	2	biimprd 154	. . . 4 |- (A.x x = y -> (ps -> ph))
9	7, 8	19.20d 996	. . 3 |- (A.x x = y -> (A.yps -> A.yph))
10		ax-10o 1140	. . . 4 |- (A.y y = x -> (A.yph -> A.xph))
11	10	alequcoms 1143	. . 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 516	1 |- (A.x x = y -> (A.xph <-> A.yps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956

This theorem is referenced by:  drex1 1156  ax11 1219  hbsb4 1248  sb9i 1263  a16g 1276  ax11indalem 1368  ax11inda2ALT 1369  ralcom2 1776  axpownd 4953

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-10 966  ax-12 968  ax-4 973  ax-5o 975  ax-10o 1140

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

Copyright terms: Public domain