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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 2079). 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 2229 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 1883). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 2112. (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 2078 | . 2 ⊢ ∃𝑥(𝜑 → 𝜑) |
3 | 2 | 19.35i 1981 | 1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1654 ∃wex 1878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-6 2075 |
This theorem depends on definitions: df-bi 199 df-ex 1879 |
This theorem is referenced by: 19.2d 2081 19.39 2087 19.24 2088 19.34 2089 eusv2i 5096 bj-ax6e 33188 bj-spnfw 33192 bj-modald 33195 wl-speqv 33852 wl-19.8eqv 33853 pm10.251 39398 ax6e2eq 39600 ax6e2eqVD 39960 |
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