Theorem sbhypf 2603

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 2560	. . 3 |- y e. _V
2		eqeq1 2046	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
3	1, 2	ceqsexv 2593	. 2 |- ( E. x ( x = y /\ x = A ) <-> y = A )
4		nfs1v 1815	. . . 4 |- F/ x [ y / x ] ph
5		sbhypf.1	. . . 4 |- F/ x ps
6	4, 5	nfbi 1481	. . 3 |- F/ x ( [ y / x ] ph <-> ps )
7		sbequ12 1654	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8	7	bicomd 129	. . . 4 |- ( x = y -> ( [ y / x ] ph <-> ph ) )
9		sbhypf.2	. . . 4 |- ( x = A -> ( ph <-> ps ) )
10	8, 9	sylan9bb 435	. . 3 |- ( ( x = y /\ x = A ) -> ( [ y / x ] ph <-> ps ) )
11	6, 10	exlimi 1485	. 2 |- ( E. x ( x = y /\ x = A ) -> ( [ y / x ] ph <-> ps ) )
12	3, 11	sylbir 125	1 |- ( y = A -> ( [ y / x ] ph <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   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-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  mob2  2721  tfisi  4310  ralxpf  4482  rexxpf  4483  nn0ind-raph  8355