Theorem bnj523

Description: Technical lemma for bnj852 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj523.1	⊢ ( 𝜑 ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj523.2	⊢ ( 𝜑′ ↔ [ 𝑀 / 𝑛 ] 𝜑 )
	bnj523.3	⊢ 𝑀 ∈ V
Assertion	bnj523	⊢ ( 𝜑′ ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj523.1	⊢ ( 𝜑 ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj523.2	⊢ ( 𝜑′ ↔ [ 𝑀 / 𝑛 ] 𝜑 )
3		bnj523.3	⊢ 𝑀 ∈ V
4	1	sbcbii	⊢ ( [ 𝑀 / 𝑛 ] 𝜑 ↔ [ 𝑀 / 𝑛 ] ( 𝐹 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
5	3	bnj525	⊢ ( [ 𝑀 / 𝑛 ] ( 𝐹 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
6	2 4 5	3bitri	⊢ ( 𝜑′ ↔ ( 𝐹 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )