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| Description: Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 1974). Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 2187 for a more conventional proof of a more general result, which uses additional axioms. The reverse implication is the defining property of effective nonfreeness (see df-nf 1783). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2006. (Revised by Wolf Lammen, 4-Dec-2017.) | 
| Ref | Expression | 
|---|---|
| 19.2 | ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . . 3 ⊢ (𝜑 → 𝜑) | |
| 2 | 1 | exgen 1973 | . 2 ⊢ ∃𝑥(𝜑 → 𝜑) | 
| 3 | 2 | 19.35i 1877 | 1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 ∃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: 19.2d 1976 19.39 1983 19.24 1984 19.34 1985 eusv2i 5393 bj-ax6e 36670 bj-spnfw 36672 bj-modald 36675 wl-speqv 37524 wl-19.8eqv 37525 pm10.251 44384 ax6e2eq 44582 ax6e2eqVD 44932 | 
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