Theorem drex1

Description: 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 drex1v if possible. (Contributed by NM, 27-Feb-2005) (New usage is discouraged.)

	Ref	Expression
Hypothesis	dral1.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	drex1	$$⊢\forall xx=y\to \left(\exists x\phi \leftrightarrow \exists y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dral1.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	notbid	$$⊢\forall xx=y\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3	2	dral1	$$⊢\forall xx=y\to \left(\forall x\neg \phi \leftrightarrow \forall y\neg \psi \right)$$
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	$$⊢\forall xx=y\to \left(\exists x\phi \leftrightarrow \exists y\psi \right)$$