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Metamath Proof Explorer
Description: Deduction form of bound-variable hypothesis builder hbim .
(Contributed by NM, 14-May-1993) (Proof shortened by Wolf Lammen, 3-Jan-2018)
|
|
Ref |
Expression |
|
Hypotheses |
hbimd.1 |
⊢ ( 𝜑 → ∀ 𝑥 𝜑 ) |
|
|
hbimd.2 |
⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 𝜓 ) ) |
|
|
hbimd.3 |
⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) ) |
|
Assertion |
hbimd |
⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → ∀ 𝑥 ( 𝜓 → 𝜒 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
hbimd.1 |
⊢ ( 𝜑 → ∀ 𝑥 𝜑 ) |
| 2 |
|
hbimd.2 |
⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 𝜓 ) ) |
| 3 |
|
hbimd.3 |
⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) ) |
| 4 |
1 2
|
nf5dh |
⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 ) |
| 5 |
1 3
|
nf5dh |
⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 ) |
| 6 |
4 5
|
nfimd |
⊢ ( 𝜑 → Ⅎ 𝑥 ( 𝜓 → 𝜒 ) ) |
| 7 |
6
|
nf5rd |
⊢ ( 𝜑 → ( ( 𝜓 → 𝜒 ) → ∀ 𝑥 ( 𝜓 → 𝜒 ) ) ) |