Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 19.21h | GIF version |
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." New proofs should use 19.21 1547 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
19.21h.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
19.21h | ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.21h.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | alim 1418 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | |
3 | 1, 2 | syl5 32 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) |
4 | hba1 1505 | . . . 4 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓) | |
5 | 1, 4 | hbim 1509 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)) |
6 | ax-4 1472 | . . . 4 ⊢ (∀𝑥𝜓 → 𝜓) | |
7 | 6 | imim2i 12 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓)) |
8 | 5, 7 | alrimih 1430 | . 2 ⊢ ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) |
9 | 3, 8 | impbii 125 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-gen 1410 ax-4 1472 ax-ial 1499 ax-i5r 1500 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: hbim1 1534 nf3 1632 19.21v 1829 |
Copyright terms: Public domain | W3C validator |