Theorem sbiev 1785

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 1784 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem sbiev

Step	Hyp	Ref	Expression
1		sbiev.1	. 2 |- F/ x ps
2		sbiev.2	. 2 |- ( x = y -> ( ph <-> ps ) )
3	1, 2	sbie 1784	1 |- ( [ y / x ] ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1453   [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-i9 1523  ax-ial 1527

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

This theorem is referenced by:  sbco2v  1941  cbvabw  2293  csbcow  3060