Metamath Proof Explorer
Description: Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011) (Revised by Mario Carneiro, 26-Jun-2015)
|
|
Ref |
Expression |
|
Hypotheses |
frins3.1 |
⊢ 𝑅 Fr 𝐴 |
|
|
frins3.2 |
⊢ 𝑅 Se 𝐴 |
|
|
frins3.3 |
⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) ) |
|
|
frins3.4 |
⊢ ( 𝑦 = 𝐵 → ( 𝜑 ↔ 𝜒 ) ) |
|
|
frins3.5 |
⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) ) |
|
Assertion |
frins3 |
⊢ ( 𝐵 ∈ 𝐴 → 𝜒 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
frins3.1 |
⊢ 𝑅 Fr 𝐴 |
2 |
|
frins3.2 |
⊢ 𝑅 Se 𝐴 |
3 |
|
frins3.3 |
⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) ) |
4 |
|
frins3.4 |
⊢ ( 𝑦 = 𝐵 → ( 𝜑 ↔ 𝜒 ) ) |
5 |
|
frins3.5 |
⊢ ( 𝑦 ∈ 𝐴 → ( ∀ 𝑧 ∈ Pred ( 𝑅 , 𝐴 , 𝑦 ) 𝜓 → 𝜑 ) ) |
6 |
1 2 3 5
|
frins2 |
⊢ ( 𝑦 ∈ 𝐴 → 𝜑 ) |
7 |
4 6
|
vtoclga |
⊢ ( 𝐵 ∈ 𝐴 → 𝜒 ) |