Theorem cbvexdw

Description: Deduction used to change bound variables, using implicit substitution. Version of cbvexd with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 2-Jan-2002) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		cbvaldw.1	$$⊢Ⅎy\phi$$
2		cbvaldw.2	$$⊢\phi \to Ⅎy\psi$$
3		cbvaldw.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	cbvaldw	$$⊢\phi \to \left(\forall x\neg \psi \leftrightarrow \forall y\neg \chi \right)$$
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	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists y\chi \right)$$