Theorem sb5f

Equivalence for substitution when \( y \) is not free in \( \varphi \). The implication "to the right" is sb1 and does not require the nonfreeness hypothesis. Theorem sb5 replaces the nonfreeness hypothesis with a disjoint variable condition on x , y and requires fewer axioms. Usage of this theorem is discouraged because it depends on [ax-13]. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)

Hypothesis
sb6f.1
\( \vdash \Finv y \, \varphi \)
Conclusion
\( \vdash [ y / x ] \, \varphi \leftrightarrow \exists x \, (x = y \wedge \varphi) \)
Proof
sb6f.1
l1
sb6f _
\( \vdash [ y / x ] \, \varphi \leftrightarrow \forall x \, (x = y \rightarrow \varphi) \)
sb6f.1
equs45f _
\( \vdash \exists x \, (x = y \wedge \varphi) \leftrightarrow \forall x \, (x = y \rightarrow \varphi) \)
bitr4i l1 _