Theorem bnj90

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Mario Carneiro, 22-Dec-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj90.1	$$⊢Y\in \mathrm{V}$$
Assertion	bnj90	$$⊢\underset{\dot{}}{[}Y/x\underset{\dot{}}{]}zFnx\leftrightarrow zFnY$$

Proof

Step	Hyp	Ref	Expression
1		bnj90.1	$$⊢Y\in \mathrm{V}$$
2		fneq2	$$⊢x=y\to \left(zFnx\leftrightarrow zFny\right)$$
3		fneq2	$$⊢y=Y\to \left(zFny\leftrightarrow zFnY\right)$$
4	2 3	sbcie2g	$$⊢Y\in \mathrm{V}\to \left(\underset{\dot{}}{[}Y/x\underset{\dot{}}{]}zFnx\leftrightarrow zFnY\right)$$
5	1 4	ax-mp	$$⊢\underset{\dot{}}{[}Y/x\underset{\dot{}}{]}zFnx\leftrightarrow zFnY$$