Theorem drnfc1

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-8 , ax-11 . (Revised by Wolf Lammen, 22-Sep-2024) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		drnfc1.1	$$⊢\forall xx=y\to A=B$$
2		eleq2w2	$$⊢A=B\to \left(w\in A\leftrightarrow w\in B\right)$$
3	1 2	syl	$$⊢\forall xx=y\to \left(w\in A\leftrightarrow w\in B\right)$$
4	3	drnf1	$$⊢\forall xx=y\to \left(Ⅎxw\in A\leftrightarrow Ⅎyw\in B\right)$$
5	4	albidv	$$⊢\forall xx=y\to \left(\forall wℲxw\in A\leftrightarrow \forall wℲyw\in B\right)$$
6		df-nfc	$$⊢\underset{\_}{Ⅎ}xA\leftrightarrow \forall wℲxw\in A$$
7		df-nfc	$$⊢\underset{\_}{Ⅎ}yB\leftrightarrow \forall wℲyw\in B$$
8	5 6 7	3bitr4g	$$⊢\forall xx=y\to \left(\underset{\_}{Ⅎ}xA\leftrightarrow \underset{\_}{Ⅎ}yB\right)$$