Theorem dral1ALT

Description: Alternate proof of dral1 , shorter but requiring ax-11 . (Contributed by NM, 24-Nov-1994) (Proof shortened by Wolf Lammen, 22-Apr-2018) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	dral1.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	dral1ALT	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		dral1.1	$$⊢\forall xx=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	dral2	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall x\psi \right)$$
3		axc11	$$⊢\forall xx=y\to \left(\forall x\psi \to \forall y\psi \right)$$
4		axc11r	$$⊢\forall xx=y\to \left(\forall y\psi \to \forall x\psi \right)$$
5	3 4	impbid	$$⊢\forall xx=y\to \left(\forall x\psi \leftrightarrow \forall y\psi \right)$$
6	2 5	bitrd	$$⊢\forall xx=y\to \left(\forall x\phi \leftrightarrow \forall y\psi \right)$$