Theorem sbhypf 2648

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)

Hypotheses

Ref	Expression
sbhypf.1	|- F/ x ps
sbhypf.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbhypf	|- ( y = A -> ( [ y / x ] ph <-> ps ) )

Distinct variable groups:   x, A   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   A( y)

Proof of Theorem sbhypf

Step	Hyp	Ref	Expression
1		vex 2604	. . 3 |- y e. _V
2		eqeq1 2087	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
3	1, 2	ceqsexv 2638	. 2 |- ( E. x ( x = y /\ x = A ) <-> y = A )
4		nfs1v 1856	. . . 4 |- F/ x [ y / x ] ph
5		sbhypf.1	. . . 4 |- F/ x ps
6	4, 5	nfbi 1521	. . 3 |- F/ x ( [ y / x ] ph <-> ps )
7		sbequ12 1694	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8	7	bicomd 139	. . . 4 |- ( x = y -> ( [ y / x ] ph <-> ph ) )
9		sbhypf.2	. . . 4 |- ( x = A -> ( ph <-> ps ) )
10	8, 9	sylan9bb 449	. . 3 |- ( ( x = y /\ x = A ) -> ( [ y / x ] ph <-> ps ) )
11	6, 10	exlimi 1525	. 2 |- ( E. x ( x = y /\ x = A ) -> ( [ y / x ] ph <-> ps ) )
12	3, 11	sylbir 133	1 |- ( y = A -> ( [ y / x ] ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  mob2  2772  tfisi  4328  ralxpf  4500  rexxpf  4501  nn0ind-raph  8464