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Mirrors > Home > MPE Home > Th. List > 19.2 | Structured version Visualization version GIF version |
Description: Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic (the other standard formulation being extru 1971). 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 2176 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 1778). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2003. (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 1970 | . 2 ⊢ ∃𝑥(𝜑 → 𝜑) |
3 | 2 | 19.35i 1873 | 1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∃wex 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-6 1963 |
This theorem depends on definitions: df-bi 206 df-ex 1774 |
This theorem is referenced by: 19.2d 1973 19.39 1980 19.24 1981 19.34 1982 eusv2i 5394 bj-ax6e 36275 bj-spnfw 36277 bj-modald 36280 wl-speqv 37120 wl-19.8eqv 37121 pm10.251 43939 ax6e2eq 44138 ax6e2eqVD 44488 |
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