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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 1475 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 1346 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | |
3 | 1, 2 | syl5 28 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) |
4 | hba1 1433 | . . . 4 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓) | |
5 | 1, 4 | hbim 1437 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)) |
6 | ax-4 1400 | . . . 4 ⊢ (∀𝑥𝜓 → 𝜓) | |
7 | 6 | imim2i 12 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓)) |
8 | 5, 7 | alrimih 1358 | . 2 ⊢ ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) |
9 | 3, 8 | impbii 117 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-4 1400 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: hbim1 1462 nf3 1559 19.21v 1753 |
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