Theorem bnj602

Description: Equality theorem for the _pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Assertion	bnj602	$$⊢X=Y\to pred\left(X,A,R\right)=pred\left(Y,A,R\right)$$

Proof

Step	Hyp	Ref	Expression
1		breq2	$$⊢X=Y\to \left(yRX\leftrightarrow yRY\right)$$
2	1	rabbidv	$$⊢X=Y\to \left\{y\in A|yRX\right\}=\left\{y\in A|yRY\right\}$$
3		df-bnj14	$$⊢pred\left(X,A,R\right)=\left\{y\in A|yRX\right\}$$
4		df-bnj14	$$⊢pred\left(Y,A,R\right)=\left\{y\in A|yRY\right\}$$
5	2 3 4	3eqtr4g	$$⊢X=Y\to pred\left(X,A,R\right)=pred\left(Y,A,R\right)$$