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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1096 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1096.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
bnj1096.2 | ⊢ (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑)) |
Ref | Expression |
---|---|
bnj1096 | ⊢ (𝜓 → ∀𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1096.2 | . 2 ⊢ (𝜓 ↔ (𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑)) | |
2 | ax-5 1827 | . . 3 ⊢ (𝜒 → ∀𝑥𝜒) | |
3 | ax-5 1827 | . . 3 ⊢ (𝜃 → ∀𝑥𝜃) | |
4 | ax-5 1827 | . . 3 ⊢ (𝜏 → ∀𝑥𝜏) | |
5 | bnj1096.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
6 | 2, 3, 4, 5 | bnj982 30103 | . 2 ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑) → ∀𝑥(𝜒 ∧ 𝜃 ∧ 𝜏 ∧ 𝜑)) |
7 | 1, 6 | hbxfrbi 1742 | 1 ⊢ (𝜓 → ∀𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∧ w-bnj17 30005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-bnj17 30006 |
This theorem is referenced by: bnj964 30267 bnj981 30274 bnj983 30275 bnj1093 30302 bnj1145 30315 |
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