Theorem drex1v

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drex1 with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 27-Feb-2005) (Revised by BJ, 17-Jun-2019)

	Ref	Expression
Hypothesis	dral1v.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	drex1v	$$⊢\forall xx=y\to \left(\exists x\phi \leftrightarrow \exists y\psi \right)$$

Proof

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