Theorem equsb1v

Substitution applied to an atomic wff. Version of equsb1 with a disjoint variable condition, which neither requires [ax-12] nor [ax-13]. (Contributed by NM, 10-May-1993.) (Revised by BJ, 11-Sep-2019.) Remove dependencies on axioms. (Revised by Wolf Lammen, 30-May-2023.) (Proof shortened by Steven Nguyen, 19-Jun-2023.) Revise [df-sb]. (Revised by Steven Nguyen, 11-Jul-2023.) (Proof shortened by Steven Nguyen, 22-Jul-2023.)

Conclusion
\( \vdash [ y' / x ] \, x = y' \)
Proof
l0
equid
\( \vdash y' = y' \)
equsb3
\( \vdash [ y' / x ] \, x = y' \leftrightarrow y' = y' \)
mpbir l0 _