Metamath Proof Explorer
Description: A lemma for eliminating a universal quantifier, in inference form.
(Contributed by Giovanni Mascellani, 30-May-2019)
|
|
Ref |
Expression |
|
Hypotheses |
spsbcdi.1 |
⊢ 𝐴 ∈ V |
|
|
spsbcdi.2 |
⊢ ( 𝜑 → ∀ 𝑥 𝜒 ) |
|
|
spsbcdi.3 |
⊢ ( [ 𝐴 / 𝑥 ] 𝜒 ↔ 𝜓 ) |
|
Assertion |
spsbcdi |
⊢ ( 𝜑 → 𝜓 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
spsbcdi.1 |
⊢ 𝐴 ∈ V |
2 |
|
spsbcdi.2 |
⊢ ( 𝜑 → ∀ 𝑥 𝜒 ) |
3 |
|
spsbcdi.3 |
⊢ ( [ 𝐴 / 𝑥 ] 𝜒 ↔ 𝜓 ) |
4 |
1
|
a1i |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
5 |
4 2
|
spsbcd |
⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜒 ) |
6 |
5 3
|
sylib |
⊢ ( 𝜑 → 𝜓 ) |