Theorem sbied 2409

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2408). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.)

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	. . . 4 |- F/ x ph
2	1	sbrim 2396	. . 3 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )
3		sbied.2	. . . . 5 |- ( ph -> F/ x ch )
4	1, 3	nfim1 2067	. . . 4 |- F/ x ( ph -> ch )
5		sbied.3	. . . . . 6 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
6	5	com12 32	. . . . 5 |- ( x = y -> ( ph -> ( ps <-> ch ) ) )
7	6	pm5.74d 262	. . . 4 |- ( x = y -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
8	4, 7	sbie 2408	. . 3 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> ch ) )
9	2, 8	bitr3i 266	. 2 |- ( ( ph -> [ y / x ] ps ) <-> ( ph -> ch ) )
10	9	pm5.74ri 261	1 |- ( ph -> ( [ y / x ] ps <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbiedv  2410  sbco2  2415  wl-equsb3  33337