Theorem sbiedh 1781

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1784). New proofs should use sbied 1782 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 1760	. . . 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		biimp 117	. . . . . . 7 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
5	3, 4	syl6 33	. . . . . 6 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
6	5	impd 252	. . . . 5 |- ( ph -> ( ( x = y /\ ps ) -> ch ) )
7	2, 6	eximdh 1605	. . . 4 |- ( ph -> ( E. x ( x = y /\ ps ) -> E. x ch ) )
8	1, 7	syl5 32	. . 3 |- ( ph -> ( [ y / x ] ps -> E. x ch ) )
9		sbiedh.2	. . . 4 |- ( ph -> ( ch -> A. x ch ) )
10	2, 9	19.9hd 1656	. . 3 |- ( ph -> ( E. x ch -> ch ) )
11	8, 10	syld 45	. 2 |- ( ph -> ( [ y / x ] ps -> ch ) )
12		biimpr 129	. . . . . . 7 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
13	3, 12	syl6 33	. . . . . 6 |- ( ph -> ( x = y -> ( ch -> ps ) ) )
14	13	com23 78	. . . . 5 |- ( ph -> ( ch -> ( x = y -> ps ) ) )
15	2, 14	alimdh 1461	. . . 4 |- ( ph -> ( A. x ch -> A. x ( x = y -> ps ) ) )
16		sb2 1761	. . . 4 |- ( A. x ( x = y -> ps ) -> [ y / x ] ps )
17	15, 16	syl6 33	. . 3 |- ( ph -> ( A. x ch -> [ y / x ] ps ) )
18	9, 17	syld 45	. 2 |- ( ph -> ( ch -> [ y / x ] ps ) )
19	11, 18	impbid 128	1 |- ( ph -> ( [ y / x ] ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1347   E.wex 1486   [wsb 1756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-4 1504  ax-i9 1524  ax-ial 1528

This theorem depends on definitions:  df-bi 116  df-sb 1757

This theorem is referenced by:  sbied  1782  sbieh  1784  sbcomxyyz  1966