Metamath Proof Explorer

Theorem drnf2

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 . Usage of nfbidv is preferred, which requires fewer axioms. (Contributed by Mario Carneiro, 4-Oct-2016) (Proof shortened by Wolf Lammen, 5-May-2018) (New usage is discouraged.)

Ref Expression
Hypothesis dral1.1 ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left({\phi }↔{\psi }\right)$
Assertion drnf2 ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left(Ⅎ{z}\phantom{\rule{.4em}{0ex}}{\phi }↔Ⅎ{z}\phantom{\rule{.4em}{0ex}}{\psi }\right)$

Proof

Step Hyp Ref Expression
1 dral1.1 ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left({\phi }↔{\psi }\right)$
2 nfae ${⊢}Ⅎ{z}\phantom{\rule{.4em}{0ex}}\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}$
3 2 1 nfbidf ${⊢}\forall {x}\phantom{\rule{.4em}{0ex}}{x}={y}\to \left(Ⅎ{z}\phantom{\rule{.4em}{0ex}}{\phi }↔Ⅎ{z}\phantom{\rule{.4em}{0ex}}{\psi }\right)$