Metamath Proof Explorer
Description: Conversion of implicit substitution to explicit class substitution,
deduction form. (Contributed by NM, 13-Dec-2014)
|
|
Ref |
Expression |
|
Hypotheses |
sbcied.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
|
|
sbcied.2 |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
|
Assertion |
sbcied |
⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
sbcied.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
sbcied.2 |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
3 |
|
nfv |
⊢ Ⅎ 𝑥 𝜑 |
4 |
|
nfvd |
⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 ) |
5 |
1 2 3 4
|
sbciedf |
⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) |