Theorem ax12ev2c

Description: A commuted form of ax12ev2 . (Contributed by BTernaryTau, 8-Sep-2026)

	Ref	Expression
Assertion	ax12ev2c	$$⊢x=y\to \left(\exists y\left(x=y\wedge \phi \right)\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		equcomi	$$⊢x=y\to y=x$$
2	1	anim1i	$$⊢\left(x=y\wedge \phi \right)\to \left(y=x\wedge \phi \right)$$
3	2	eximi	$$⊢\exists y\left(x=y\wedge \phi \right)\to \exists y\left(y=x\wedge \phi \right)$$
4		ax12ev2	$$⊢\exists y\left(y=x\wedge \phi \right)\to \left(y=x\to \phi \right)$$
5	3 1 4	syl2imc	$$⊢x=y\to \left(\exists y\left(x=y\wedge \phi \right)\to \phi \right)$$