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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1275 | 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 |
---|---|
bnj1275.1 | ⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒)) |
bnj1275.2 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
bnj1275 | ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1275.2 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | bnj1275.1 | . . 3 ⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒)) | |
3 | 1, 2 | bnj596 31916 | . 2 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ (𝜓 ∧ 𝜒))) |
4 | 3anass 1087 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒))) | |
5 | 3, 4 | bnj1198 31966 | 1 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 ∀wal 1526 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-3an 1081 df-ex 1772 df-nf 1776 |
This theorem is referenced by: bnj1345 31995 bnj1279 32187 |
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