Theorem sbiedv 1812

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1814). (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 1551	. 2 |- F/ x ph
2		nfvd 1552	. 2 |- ( ph -> F/ x ch )
3		sbiedv.1	. . 3 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
4	3	ex 115	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5	1, 2, 4	sbied 1811	1 |- ( ph -> ( [ y / x ] ps <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   [wsb 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by:  acexmid  5945