Theorem bnj91

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj91.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj91.2	⊢ 𝑍 ∈ V
Assertion	bnj91	⊢ ( [ 𝑍 / 𝑦 ] 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj91.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
2		bnj91.2	⊢ 𝑍 ∈ V
3	1	sbcbii	⊢ ( [ 𝑍 / 𝑦 ] 𝜑 ↔ [ 𝑍 / 𝑦 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
4	2	bnj525	⊢ ( [ 𝑍 / 𝑦 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
5	3 4	bitri	⊢ ( [ 𝑍 / 𝑦 ] 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )