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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj982 | 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 |
---|---|
bnj982.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
bnj982.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
bnj982.3 | ⊢ (𝜒 → ∀𝑥𝜒) |
bnj982.4 | ⊢ (𝜃 → ∀𝑥𝜃) |
Ref | Expression |
---|---|
bnj982 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bnj17 31952 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) | |
2 | bnj982.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | bnj982.2 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | bnj982.3 | . . . 4 ⊢ (𝜒 → ∀𝑥𝜒) | |
5 | 2, 3, 4 | hb3an 2305 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) |
6 | bnj982.4 | . . 3 ⊢ (𝜃 → ∀𝑥𝜃) | |
7 | 5, 6 | hban 2304 | . 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → ∀𝑥((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) |
8 | 1, 7 | hbxfrbi 1821 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∀wal 1531 ∧ w-bnj17 31951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-12 2173 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-bnj17 31952 |
This theorem is referenced by: bnj1096 32049 bnj1311 32291 bnj1445 32311 |
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