Theorem sbiedh 1670

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1673). New proofs should use sbied 1671 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem sbiedh

Step	Hyp	Ref	Expression
1		sb1 1649	. . . 4 |- ( [ y / x ] ps -> E. x ( x = y /\ ps ) )
2		sbiedh.1	. . . . 5 |- ( ph -> A. x ph )
3		sbiedh.3	. . . . . . 7 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
4		bi1 111	. . . . . . 7 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
5	3, 4	syl6 29	. . . . . 6 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
6	5	impd 242	. . . . 5 |- ( ph -> ( ( x = y /\ ps ) -> ch ) )
7	2, 6	eximdh 1502	. . . 4 |- ( ph -> ( E. x ( x = y /\ ps ) -> E. x ch ) )
8	1, 7	syl5 28	. . 3 |- ( ph -> ( [ y / x ] ps -> E. x ch ) )
9		sbiedh.2	. . . 4 |- ( ph -> ( ch -> A. x ch ) )
10	2, 9	19.9hd 1552	. . 3 |- ( ph -> ( E. x ch -> ch ) )
11	8, 10	syld 40	. 2 |- ( ph -> ( [ y / x ] ps -> ch ) )
12		bi2 121	. . . . . . 7 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
13	3, 12	syl6 29	. . . . . 6 |- ( ph -> ( x = y -> ( ch -> ps ) ) )
14	13	com23 72	. . . . 5 |- ( ph -> ( ch -> ( x = y -> ps ) ) )
15	2, 14	alimdh 1356	. . . 4 |- ( ph -> ( A. x ch -> A. x ( x = y -> ps ) ) )
16		sb2 1650	. . . 4 |- ( A. x ( x = y -> ps ) -> [ y / x ] ps )
17	15, 16	syl6 29	. . 3 |- ( ph -> ( A. x ch -> [ y / x ] ps ) )
18	9, 17	syld 40	. 2 |- ( ph -> ( ch -> [ y / x ] ps ) )
19	11, 18	impbid 120	1 |- ( ph -> ( [ y / x ] ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  sbied  1671  sbieh  1673  sbcomxyyz  1846