Theorem drnf1

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016) Usage of this theorem is discouraged because it depends on ax-13 . Use drnf1v instead. (New usage is discouraged.)

	Ref	Expression
Hypothesis	dral1.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	drnf1	$$⊢\forall xx=y\to \left(Ⅎx\phi \leftrightarrow Ⅎy\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dral1.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	dral1	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall y\psi \right)$$
3	1 2	imbi12d	$$⊢\forall xx=y\to \left(\left(\phi \to \forall x\phi \right)\leftrightarrow \left(\psi \to \forall y\psi \right)\right)$$
4	3	dral1	$$⊢\forall xx=y\to \left(\forall x\left(\phi \to \forall x\phi \right)\leftrightarrow \forall y\left(\psi \to \forall y\psi \right)\right)$$
5		nf5	$$⊢Ⅎx\phi \leftrightarrow \forall x\left(\phi \to \forall x\phi \right)$$
6		nf5	$$⊢Ⅎy\psi \leftrightarrow \forall y\left(\psi \to \forall y\psi \right)$$
7	4 5 6	3bitr4g	$$⊢\forall xx=y\to \left(Ⅎx\phi \leftrightarrow Ⅎy\psi \right)$$