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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1198 | 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 | 
|---|---|
| bnj1198.1 | ⊢ (𝜑 → ∃𝑥𝜓) | 
| bnj1198.2 | ⊢ (𝜓′ ↔ 𝜓) | 
| Ref | Expression | 
|---|---|
| bnj1198 | ⊢ (𝜑 → ∃𝑥𝜓′) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj1198.1 | . 2 ⊢ (𝜑 → ∃𝑥𝜓) | |
| 2 | bnj1198.2 | . . 3 ⊢ (𝜓′ ↔ 𝜓) | |
| 3 | 2 | exbii 1848 | . 2 ⊢ (∃𝑥𝜓′ ↔ ∃𝑥𝜓) | 
| 4 | 1, 3 | sylibr 234 | 1 ⊢ (𝜑 → ∃𝑥𝜓′) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∃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 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 | 
| This theorem is referenced by: bnj1209 34810 bnj1275 34827 bnj1340 34837 bnj1345 34838 bnj605 34921 bnj607 34930 bnj906 34944 bnj908 34945 bnj1189 35023 bnj1450 35064 bnj1312 35072 | 
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