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| Description: Rule of existential generalization, similar to universal generalization ax-gen 1794, but valid only if an individual exists. Its proof requires ax-6 1966 in our axiomatization but the equality predicate does not occur in its statement. Some fundamental theorems of predicate calculus can be proven from ax-gen 1794, ax-4 1808 and this theorem alone, not requiring ax-7 2006 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.) | 
| Ref | Expression | 
|---|---|
| exgen.1 | ⊢ 𝜑 | 
| Ref | Expression | 
|---|---|
| exgen | ⊢ ∃𝑥𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idd 24 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜑)) | |
| 2 | exgen.1 | . 2 ⊢ 𝜑 | |
| 3 | 1, 2 | speiv 1971 | 1 ⊢ ∃𝑥𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∃wex 1778 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-6 1966 | 
| This theorem depends on definitions: df-bi 207 df-ex 1779 | 
| This theorem is referenced by: extru 1974 19.2 1975 vn0 4344 axnulg 35107 ac6s6 38180 | 
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