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Metamath Proof Explorer

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 ) ) )