Theorem bnj1040

Description: Technical lemma for bnj69 . 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	bnj1040.1	⊢ ( 𝜑′ ↔ [ 𝑗 / 𝑖 ] 𝜑 )
	bnj1040.2	⊢ ( 𝜓′ ↔ [ 𝑗 / 𝑖 ] 𝜓 )
	bnj1040.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj1040.4	⊢ ( 𝜒′ ↔ [ 𝑗 / 𝑖 ] 𝜒 )
Assertion	bnj1040	⊢ ( 𝜒′ ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1040.1	⊢ ( 𝜑′ ↔ [ 𝑗 / 𝑖 ] 𝜑 )
2		bnj1040.2	⊢ ( 𝜓′ ↔ [ 𝑗 / 𝑖 ] 𝜓 )
3		bnj1040.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj1040.4	⊢ ( 𝜒′ ↔ [ 𝑗 / 𝑖 ] 𝜒 )
5	3	sbcbii	⊢ ( [ 𝑗 / 𝑖 ] 𝜒 ↔ [ 𝑗 / 𝑖 ] ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
6		df-bnj17	⊢ ( ( [ 𝑗 / 𝑖 ] 𝑛 ∈ 𝐷 ∧ [ 𝑗 / 𝑖 ] 𝑓 Fn 𝑛 ∧ [ 𝑗 / 𝑖 ] 𝜑 ∧ [ 𝑗 / 𝑖 ] 𝜓 ) ↔ ( ( [ 𝑗 / 𝑖 ] 𝑛 ∈ 𝐷 ∧ [ 𝑗 / 𝑖 ] 𝑓 Fn 𝑛 ∧ [ 𝑗 / 𝑖 ] 𝜑 ) ∧ [ 𝑗 / 𝑖 ] 𝜓 ) )
7		vex	⊢ 𝑗 ∈ V
8	7	bnj525	⊢ ( [ 𝑗 / 𝑖 ] 𝑛 ∈ 𝐷 ↔ 𝑛 ∈ 𝐷 )
9	8	bicomi	⊢ ( 𝑛 ∈ 𝐷 ↔ [ 𝑗 / 𝑖 ] 𝑛 ∈ 𝐷 )
10	7	bnj525	⊢ ( [ 𝑗 / 𝑖 ] 𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛 )
11	10	bicomi	⊢ ( 𝑓 Fn 𝑛 ↔ [ 𝑗 / 𝑖 ] 𝑓 Fn 𝑛 )
12	9 11 1 2	bnj887	⊢ ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( [ 𝑗 / 𝑖 ] 𝑛 ∈ 𝐷 ∧ [ 𝑗 / 𝑖 ] 𝑓 Fn 𝑛 ∧ [ 𝑗 / 𝑖 ] 𝜑 ∧ [ 𝑗 / 𝑖 ] 𝜓 ) )
13		df-bnj17	⊢ ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ) ∧ 𝜓 ) )
14	13	sbcbii	⊢ ( [ 𝑗 / 𝑖 ] ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ [ 𝑗 / 𝑖 ] ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ) ∧ 𝜓 ) )
15		sbcan	⊢ ( [ 𝑗 / 𝑖 ] ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ) ∧ 𝜓 ) ↔ ( [ 𝑗 / 𝑖 ] ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ) ∧ [ 𝑗 / 𝑖 ] 𝜓 ) )
16		sbc3an	⊢ ( [ 𝑗 / 𝑖 ] ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ) ↔ ( [ 𝑗 / 𝑖 ] 𝑛 ∈ 𝐷 ∧ [ 𝑗 / 𝑖 ] 𝑓 Fn 𝑛 ∧ [ 𝑗 / 𝑖 ] 𝜑 ) )
17	16	anbi1i	⊢ ( ( [ 𝑗 / 𝑖 ] ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ) ∧ [ 𝑗 / 𝑖 ] 𝜓 ) ↔ ( ( [ 𝑗 / 𝑖 ] 𝑛 ∈ 𝐷 ∧ [ 𝑗 / 𝑖 ] 𝑓 Fn 𝑛 ∧ [ 𝑗 / 𝑖 ] 𝜑 ) ∧ [ 𝑗 / 𝑖 ] 𝜓 ) )
18	14 15 17	3bitri	⊢ ( [ 𝑗 / 𝑖 ] ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( ( [ 𝑗 / 𝑖 ] 𝑛 ∈ 𝐷 ∧ [ 𝑗 / 𝑖 ] 𝑓 Fn 𝑛 ∧ [ 𝑗 / 𝑖 ] 𝜑 ) ∧ [ 𝑗 / 𝑖 ] 𝜓 ) )
19	6 12 18	3bitr4ri	⊢ ( [ 𝑗 / 𝑖 ] ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) )
20	4 5 19	3bitri	⊢ ( 𝜒′ ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′ ) )