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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 34808 | . 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 34838 | 1 ⊢ (𝜒 → ∃𝑥𝜃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-ex 1780 df-nf 1784 df-rex 3071 |
| This theorem is referenced by: bnj1204 35026 bnj1311 35038 bnj1398 35048 bnj1408 35050 bnj1450 35064 bnj1312 35072 bnj1501 35081 bnj1523 35085 |
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