Theorem cbvexd

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvexdw if possible. (Contributed by NM, 2-Jan-2002) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvald.1	$$⊢Ⅎy\phi$$
	cbvald.2	$$⊢\phi \to Ⅎy\psi$$
	cbvald.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	cbvexd	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvald.1	$$⊢Ⅎy\phi$$
2		cbvald.2	$$⊢\phi \to Ⅎy\psi$$
3		cbvald.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
4	2	nfnd	$$⊢\phi \to Ⅎy\neg \psi$$
5		notbi	$$⊢\left(\psi \leftrightarrow \chi \right)\leftrightarrow \left(\neg \psi \leftrightarrow \neg \chi \right)$$
6	3 5	imbitrdi	$$⊢\phi \to \left(x=y\to \left(\neg \psi \leftrightarrow \neg \chi \right)\right)$$
7	1 4 6	cbvald	$$⊢\phi \to \left(\forall x\neg \psi \leftrightarrow \forall y\neg \chi \right)$$
8		alnex	$$⊢\forall x\neg \psi \leftrightarrow \neg \exists x\psi$$
9		alnex	$$⊢\forall y\neg \chi \leftrightarrow \neg \exists y\chi$$
10	7 8 9	3bitr3g	$$⊢\phi \to \left(\neg \exists x\psi \leftrightarrow \neg \exists y\chi \right)$$
11	10	con4bid	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists y\chi \right)$$