Theorem dral1

Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on [ax-13]. Use the weaker dral1v if possible. (Contributed by NM, 24-Nov-1994.) Remove dependency on [ax-11]. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)

Hypothesis
dral1.1
\( \vdash \forall x \, x = y \rightarrow (\varphi \leftrightarrow \psi) \)
Conclusion
\( \vdash \forall x \, x = y \rightarrow (\forall x \, \varphi \leftrightarrow \forall y \, \psi) \)
Proof
l0
nfa1
dral1.1
l2
albid l0 _
\( \vdash \forall x \, x = y \rightarrow (\forall x \, \varphi \leftrightarrow \forall x \, \psi) \)
l3
axc11
axc11r
impbid l3 _
\( \vdash \forall x \, x = y \rightarrow (\forall x \, \psi \leftrightarrow \forall y \, \psi) \)
bitrd l2 _