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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1131 | 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 | 
|---|---|
| bnj1131.1 | ⊢ (𝜑 → ∀𝑥𝜑) | 
| bnj1131.2 | ⊢ ∃𝑥𝜑 | 
| Ref | Expression | 
|---|---|
| bnj1131 | ⊢ 𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj1131.2 | . 2 ⊢ ∃𝑥𝜑 | |
| 2 | bnj1131.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
| 3 | 2 | 19.9h 2286 | . 2 ⊢ (∃𝑥𝜑 ↔ 𝜑) | 
| 4 | 1, 3 | mpbi 230 | 1 ⊢ 𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 | 
| 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-ex 1780 df-nf 1784 | 
| This theorem is referenced by: bnj1468 34860 bnj1014 34975 bnj1128 35004 | 
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