Theorem bnj121

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

	Ref	Expression
Hypotheses	bnj121.1	⊢ ( 𝜁 ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
	bnj121.2	⊢ ( 𝜁′ ↔ [ 1o / 𝑛 ] 𝜁 )
	bnj121.3	⊢ ( 𝜑′ ↔ [ 1o / 𝑛 ] 𝜑 )
	bnj121.4	⊢ ( 𝜓′ ↔ [ 1o / 𝑛 ] 𝜓 )
Assertion	bnj121	⊢ ( 𝜁′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj121.1	⊢ ( 𝜁 ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
2		bnj121.2	⊢ ( 𝜁′ ↔ [ 1o / 𝑛 ] 𝜁 )
3		bnj121.3	⊢ ( 𝜑′ ↔ [ 1o / 𝑛 ] 𝜑 )
4		bnj121.4	⊢ ( 𝜓′ ↔ [ 1o / 𝑛 ] 𝜓 )
5	1	sbcbii	⊢ ( [ 1o / 𝑛 ] 𝜁 ↔ [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
6		bnj105	⊢ 1o ∈ V
7	6	bnj90	⊢ ( [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ↔ 𝑓 Fn 1o )
8	7	bicomi	⊢ ( 𝑓 Fn 1o ↔ [ 1o / 𝑛 ] 𝑓 Fn 𝑛 )
9	8 3 4	3anbi123i	⊢ ( ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ ( [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ∧ [ 1o / 𝑛 ] 𝜑 ∧ [ 1o / 𝑛 ] 𝜓 ) )
10		sbc3an	⊢ ( [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( [ 1o / 𝑛 ] 𝑓 Fn 𝑛 ∧ [ 1o / 𝑛 ] 𝜑 ∧ [ 1o / 𝑛 ] 𝜓 ) )
11	9 10	bitr4i	⊢ ( ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ↔ [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
12	11	imbi2i	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
13		nfv	⊢ Ⅎ 𝑛 ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 )
14	13	sbc19.21g	⊢ ( 1o ∈ V → ( [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ) )
15	6 14	ax-mp	⊢ ( [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → [ 1o / 𝑛 ] ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
16	12 15	bitr4i	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) ↔ [ 1o / 𝑛 ] ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
17	5 2 16	3bitr4i	⊢ ( 𝜁′ ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′ ) ) )