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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 2028 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
2 | 1 | nf5ri 2065 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-or 385 df-ex 1705 df-nf 1710 |
This theorem is referenced by: nfa1OLD 2157 nfaldOLD 2166 nfa1OLDOLD 2207 axi5r 2594 axial 2595 bj-19.41al 32637 bj-modal4e 32705 hbntal 38769 hbimpg 38770 hbimpgVD 39140 hbalgVD 39141 hbexgVD 39142 ax6e2eqVD 39143 e2ebindVD 39148 vk15.4jVD 39150 |
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