Theorem sbiev 1803

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 1802 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 1802	1 |- ( [ y / x ] ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1471   [wsb 1773

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sbco2v  1960  cbvabw  2312  csbcow  3083