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Metamath Proof Explorer

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′ ) ) )