Theorem sbied 1811

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1814). (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 1542	. 2 |- ( ph -> A. x ph )
3		sbied.2	. . 3 |- ( ph -> F/ x ch )
4	3	nfrd 1543	. 2 |- ( ph -> ( ch -> A. x ch ) )
5		sbied.3	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
6	2, 4, 5	sbiedh 1810	1 |- ( ph -> ( [ y / x ] ps <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1483   [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-i9 1553  ax-ial 1557

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

This theorem is referenced by:  sbiedv  1812  dvelimdf  2044  cbvrald  15761