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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 33805 | . 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 33835 | 1 ⊢ (𝜒 → ∃𝑥𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 ∀wal 1540 ∃wex 1782 ∈ wcel 2107 ∃wrex 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-ex 1783 df-nf 1787 df-rex 3072 |
This theorem is referenced by: bnj1204 34023 bnj1311 34035 bnj1398 34045 bnj1408 34047 bnj1450 34061 bnj1312 34069 bnj1501 34078 bnj1523 34082 |
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