Theorem bnj591

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
Hypothesis	bnj591.1	⊢ ( 𝜃 ↔ ( ( 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′ ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ) )
Assertion	bnj591	⊢ ( [ 𝑘 / 𝑗 ] 𝜃 ↔ ( ( 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′ ) → ( 𝑓 ‘ 𝑘 ) = ( 𝑔 ‘ 𝑘 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj591.1	⊢ ( 𝜃 ↔ ( ( 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′ ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ) )
2	1	sbcbii	⊢ ( [ 𝑘 / 𝑗 ] 𝜃 ↔ [ 𝑘 / 𝑗 ] ( ( 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′ ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ) )
3		vex	⊢ 𝑘 ∈ V
4		fveq2	⊢ ( 𝑗 = 𝑘 → ( 𝑓 ‘ 𝑗 ) = ( 𝑓 ‘ 𝑘 ) )
5		fveq2	⊢ ( 𝑗 = 𝑘 → ( 𝑔 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑘 ) )
6	4 5	eqeq12d	⊢ ( 𝑗 = 𝑘 → ( ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ↔ ( 𝑓 ‘ 𝑘 ) = ( 𝑔 ‘ 𝑘 ) ) )
7	6	imbi2d	⊢ ( 𝑗 = 𝑘 → ( ( ( 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′ ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ) ↔ ( ( 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′ ) → ( 𝑓 ‘ 𝑘 ) = ( 𝑔 ‘ 𝑘 ) ) ) )
8	3 7	sbcie	⊢ ( [ 𝑘 / 𝑗 ] ( ( 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′ ) → ( 𝑓 ‘ 𝑗 ) = ( 𝑔 ‘ 𝑗 ) ) ↔ ( ( 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′ ) → ( 𝑓 ‘ 𝑘 ) = ( 𝑔 ‘ 𝑘 ) ) )
9	2 8	bitri	⊢ ( [ 𝑘 / 𝑗 ] 𝜃 ↔ ( ( 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′ ) → ( 𝑓 ‘ 𝑘 ) = ( 𝑔 ‘ 𝑘 ) ) )