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 ∈ V
Assertion	bnj90	⊢ ( [ 𝑌 / 𝑥 ] 𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		bnj90.1	⊢ 𝑌 ∈ V
2		fneq2	⊢ ( 𝑥 = 𝑦 → ( 𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑦 ) )
3		fneq2	⊢ ( 𝑦 = 𝑌 → ( 𝑧 Fn 𝑦 ↔ 𝑧 Fn 𝑌 ) )
4	2 3	sbcie2g	⊢ ( 𝑌 ∈ V → ( [ 𝑌 / 𝑥 ] 𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌 ) )
5	1 4	ax-mp	⊢ ( [ 𝑌 / 𝑥 ] 𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌 )