Metamath Proof Explorer
Description: _E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011) (Revised by Mario Carneiro, 11-Dec-2016)
|
|
Ref |
Expression |
|
Hypotheses |
setinds2f.1 |
⊢ Ⅎ 𝑥 𝜓 |
|
|
setinds2f.2 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) |
|
|
setinds2f.3 |
⊢ ( ∀ 𝑦 ∈ 𝑥 𝜓 → 𝜑 ) |
|
Assertion |
setinds2f |
⊢ 𝜑 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
setinds2f.1 |
⊢ Ⅎ 𝑥 𝜓 |
2 |
|
setinds2f.2 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) |
3 |
|
setinds2f.3 |
⊢ ( ∀ 𝑦 ∈ 𝑥 𝜓 → 𝜑 ) |
4 |
|
sbsbc |
⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) |
5 |
1 2
|
sbiev |
⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) |
6 |
4 5
|
bitr3i |
⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) |
7 |
6
|
ralbii |
⊢ ( ∀ 𝑦 ∈ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ∈ 𝑥 𝜓 ) |
8 |
7 3
|
sylbi |
⊢ ( ∀ 𝑦 ∈ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 → 𝜑 ) |
9 |
8
|
setinds |
⊢ 𝜑 |