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Mirrors > Home > MPE Home > Th. List > hba1 | Structured version Visualization version GIF version |
Description: The setvar 𝑥 is not free in ∀𝑥𝜑. This corresponds to the axiom (4) of modal logic. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 12-Oct-2021.) |
Ref | Expression |
---|---|
hba1 | ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2068 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
2 | 1 | nf5ri 2103 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-10 2059 ax-12 2087 |
This theorem depends on definitions: df-bi 197 df-or 384 df-ex 1745 df-nf 1750 |
This theorem is referenced by: nfa1OLD 2195 nfaldOLD 2202 nfa1OLDOLD 2243 axi5r 2623 axial 2624 bj-19.41al 32762 bj-modal4e 32830 hbntal 39086 hbimpg 39087 hbimpgVD 39454 hbalgVD 39455 hbexgVD 39456 ax6e2eqVD 39457 e2ebindVD 39462 vk15.4jVD 39464 |
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