Theorem drnf1v

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 4-Oct-2016) (Revised by BJ, 17-Jun-2019) Avoid ax-10 . (Revised by Gino Giotto, 18-Nov-2024)

	Ref	Expression
Hypothesis	dral1v.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	drnf1v	$$⊢\forall xx=y\to \left(Ⅎx\phi \leftrightarrow Ⅎy\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dral1v.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	drex1v	$$⊢\forall xx=y\to \left(\exists x\phi \leftrightarrow \exists y\psi \right)$$
3	1	dral1v	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall y\psi \right)$$
4	2 3	imbi12d	$$⊢\forall xx=y\to \left(\left(\exists x\phi \to \forall x\phi \right)\leftrightarrow \left(\exists y\psi \to \forall y\psi \right)\right)$$
5		df-nf	$$⊢Ⅎx\phi \leftrightarrow \left(\exists x\phi \to \forall x\phi \right)$$
6		df-nf	$$⊢Ⅎy\psi \leftrightarrow \left(\exists y\psi \to \forall y\psi \right)$$
7	4 5 6	3bitr4g	$$⊢\forall xx=y\to \left(Ⅎx\phi \leftrightarrow Ⅎy\psi \right)$$