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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1521 | 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 |
---|---|
bnj1521.1 | ⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑) |
bnj1521.2 | ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑)) |
bnj1521.3 | ⊢ (𝜒 → ∀𝑥𝜒) |
Ref | Expression |
---|---|
bnj1521 | ⊢ (𝜒 → ∃𝑥𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1521.1 | . . 3 ⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑) | |
2 | 1 | bnj1196 34787 | . 2 ⊢ (𝜒 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) |
3 | bnj1521.2 | . 2 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑)) | |
4 | bnj1521.3 | . 2 ⊢ (𝜒 → ∀𝑥𝜒) | |
5 | 2, 3, 4 | bnj1345 34817 | 1 ⊢ (𝜒 → ∃𝑥𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 ∀wal 1535 ∃wex 1776 ∈ wcel 2106 ∃wrex 3068 |
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 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-ex 1777 df-nf 1781 df-rex 3069 |
This theorem is referenced by: bnj1204 35005 bnj1311 35017 bnj1398 35027 bnj1408 35029 bnj1450 35043 bnj1312 35051 bnj1501 35060 bnj1523 35064 |
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