| Mathbox for Jonathan Ben-Naim |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj31 | 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 |
|---|---|
| bnj31.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
| bnj31.2 | ⊢ (𝜓 → 𝜒) |
| Ref | Expression |
|---|---|
| bnj31 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj31.1 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | |
| 2 | bnj31.2 | . . 3 ⊢ (𝜓 → 𝜒) | |
| 3 | 2 | reximi 3084 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒) |
| 4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∃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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-rex 3071 |
| This theorem is referenced by: bnj168 34744 bnj110 34872 bnj906 34944 bnj1253 35031 bnj1280 35034 bnj1296 35035 bnj1371 35043 bnj1497 35074 bnj1498 35075 bnj1501 35081 |
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