Theorem sbied 1776

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1779). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypotheses

Ref	Expression
sbied.1	|- F/ x ph
sbied.2	|- ( ph -> F/ x ch )
sbied.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

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

Proof of Theorem sbied

Step	Hyp	Ref	Expression
1		sbied.1	. . 3 |- F/ x ph
2	1	nfri 1507	. 2 |- ( ph -> A. x ph )
3		sbied.2	. . 3 |- ( ph -> F/ x ch )
4	3	nfrd 1508	. 2 |- ( ph -> ( ch -> A. x ch ) )
5		sbied.3	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
6	2, 4, 5	sbiedh 1775	1 |- ( ph -> ( [ y / x ] ps <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1448   [wsb 1750

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sbiedv  1777  dvelimdf  2004  cbvrald  13669