Theorem sbiedv 1782

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1784). (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 1521	. 2 |- F/ x ph
2		nfvd 1522	. 2 |- ( ph -> F/ x ch )
3		sbiedv.1	. . 3 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
4	3	ex 114	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5	1, 2, 4	sbied 1781	1 |- ( ph -> ( [ y / x ] ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   [wsb 1755

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  acexmid  5852