Theorem sbiedOLD 1564

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1565).

Hypotheses

Ref	Expression
sbied.1	|- (ph -> A.xph)
sbied.2	|- (ph -> (ch -> A.xch))
sbied.3	|- (ph -> (x = y -> (ps <-> ch)))

Assertion

Ref	Expression
sbiedOLD	|- (ph -> ([y / x]ps <-> ch))

Proof of Theorem sbiedOLD

Step	Hyp	Ref	Expression
1		sbied.1	. . 3 |- (ph -> A.xph)
2		sbied.3	. . . . . . . . 9 |- (ph -> (x = y -> (ps <-> ch)))
3		bi1 165	. . . . . . . . 9 |- ((ps <-> ch) -> (ps -> ch))
4	2, 3	syl6 25	. . . . . . . 8 |- (ph -> (x = y -> (ps -> ch)))
5	4	imp3a 388	. . . . . . 7 |- (ph -> ((x = y /\ ps) -> ch))
6	5	alimi 1338	. . . . . 6 |- (A.xph -> A.x((x = y /\ ps) -> ch))
7		exim 1386	. . . . . 6 |- (A.x((x = y /\ ps) -> ch) -> (E.x(x = y /\ ps) -> E.xch))
8	6, 7	syl 12	. . . . 5 |- (A.xph -> (E.x(x = y /\ ps) -> E.xch))
9		sb1 1540	. . . . 5 |- ([y / x]ps -> E.x(x = y /\ ps))
10	8, 9	syl5 20	. . . 4 |- (A.xph -> ([y / x]ps -> E.xch))
11		sbied.2	. . . . . . 7 |- (ph -> (ch -> A.xch))
12	11	alimi 1338	. . . . . 6 |- (A.xph -> A.x(ch -> A.xch))
13		hba1 1350	. . . . . . 7 |- (A.xch -> A.xA.xch)
14	13	19.23 1411	. . . . . 6 |- (A.x(ch -> A.xch) <-> (E.xch -> A.xch))
15	12, 14	sylib 215	. . . . 5 |- (A.xph -> (E.xch -> A.xch))
16		ax-4 1319	. . . . 5 |- (A.xch -> ch)
17	15, 16	syl6 25	. . . 4 |- (A.xph -> (E.xch -> ch))
18	10, 17	syld 30	. . 3 |- (A.xph -> ([y / x]ps -> ch))
19	1, 18	syl 12	. 2 |- (ph -> ([y / x]ps -> ch))
20		bi2 166	. . . . . . 7 |- ((ps <-> ch) -> (ch -> ps))
21	2, 20	syl6 25	. . . . . 6 |- (ph -> (x = y -> (ch -> ps)))
22	21	com23 36	. . . . 5 |- (ph -> (ch -> (x = y -> ps)))
23	22	al2imi 1341	. . . 4 |- (A.xph -> (A.xch -> A.x(x = y -> ps)))
24		sb2 1541	. . . 4 |- (A.x(x = y -> ps) -> [y / x]ps)
25	23, 24	syl6 25	. . 3 |- (A.xph -> (A.xch -> [y / x]ps))
26	1, 11, 25	sylsyld 32	. 2 |- (ph -> (ch -> [y / x]ps))
27	19, 26	impbid 574	1 |- (ph -> ([y / x]ps <-> ch))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326  [wsbc 1534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536

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