Theorem drnfc2

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv with dral2 , leading to a one byte longer proof. However feel free to manually edit it according to conventions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 8-Oct-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	drnfc1.1	$$⊢\forall xx=y\to A=B$$
Assertion	drnfc2	$$⊢\forall xx=y\to \left(\underset{\_}{Ⅎ}zA\leftrightarrow \underset{\_}{Ⅎ}zB\right)$$

Proof

Step	Hyp	Ref	Expression
1		drnfc1.1	$$⊢\forall xx=y\to A=B$$
2	1	eleq2d	$$⊢\forall xx=y\to \left(w\in A\leftrightarrow w\in B\right)$$
3	2	drnf2	$$⊢\forall xx=y\to \left(Ⅎzw\in A\leftrightarrow Ⅎzw\in B\right)$$
4	3	albidv	$$⊢\forall xx=y\to \left(\forall wℲzw\in A\leftrightarrow \forall wℲzw\in B\right)$$
5		df-nfc	$$⊢\underset{\_}{Ⅎ}zA\leftrightarrow \forall wℲzw\in A$$
6		df-nfc	$$⊢\underset{\_}{Ⅎ}zB\leftrightarrow \forall wℲzw\in B$$
7	4 5 6	3bitr4g	$$⊢\forall xx=y\to \left(\underset{\_}{Ⅎ}zA\leftrightarrow \underset{\_}{Ⅎ}zB\right)$$