Theorem sbiedv 1672

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1674). (Contributed by NM, 7-Jan-2017.)

Hypothesis

Ref	Expression
sbiedv.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )

Assertion

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

Distinct variable groups:   ph, x   ch, x

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

Proof of Theorem sbiedv

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 |- F/ x ph
2		nfvd 1422	. 2 |- ( ph -> F/ x ch )
3		sbiedv.1	. . 3 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
4	3	ex 108	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5	1, 2, 4	sbied 1671	1 |- ( ph -> ( [ y / x ] ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   [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-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  acexmid  5511