Theorem dral1v

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 with a disjoint variable condition, which does not require ax-13 . Remark: the corresponding versions for dral2 and drex2 are instances of albidv and exbidv respectively. (Contributed by NM, 24-Nov-1994) (Revised by BJ, 17-Jun-2019) Base the proof on ax12v . (Revised by Wolf Lammen, 30-Mar-2024)

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

Proof

Step	Hyp	Ref	Expression
1		dral1v.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
2		nfa1	$$⊢Ⅎx\forall xx=y$$
3	2 1	albid	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall x\psi \right)$$
4		axc11v	$$⊢\forall xx=y\to \left(\forall x\psi \to \forall y\psi \right)$$
5		axc11rv	$$⊢\forall xx=y\to \left(\forall y\psi \to \forall x\psi \right)$$
6	4 5	impbid	$$⊢\forall xx=y\to \left(\forall x\psi \leftrightarrow \forall y\psi \right)$$
7	3 6	bitrd	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall y\psi \right)$$