Theorem abbi1sn

Description: Originally part of uniabio . Convert a theorem about df-iota to one about dfiota2 , without ax-10 , ax-11 , ax-12 . Although, eu6 uses ax-10 and ax-12 . (Contributed by SN, 23-Nov-2024)

	Ref	Expression
Assertion	abbi1sn	$$⊢\forall x\left(\phi \leftrightarrow x=y\right)\to \left\{x|\phi \right\}=\left\{y\right\}$$

Proof

Step	Hyp	Ref	Expression
1		abbi	$$⊢\forall x\left(\phi \leftrightarrow x=y\right)\to \left\{x|\phi \right\}=\left\{x|x=y\right\}$$
2		df-sn	$$⊢\left\{y\right\}=\left\{x|x=y\right\}$$
3	1 2	eqtr4di	$$⊢\forall x\left(\phi \leftrightarrow x=y\right)\to \left\{x|\phi \right\}=\left\{y\right\}$$