Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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, 12-Oct-2021.) |
Ref | Expression |
---|---|
hba1 | ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2151 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
2 | 1 | nf5ri 2191 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-12 2173 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1777 df-nf 1781 |
This theorem is referenced by: axi5r 2783 axial 2784 bj-19.41al 33987 bj-wnf1 34046 hbntal 40880 hbimpg 40881 hbimpgVD 41231 hbalgVD 41232 hbexgVD 41233 ax6e2eqVD 41234 e2ebindVD 41239 vk15.4jVD 41241 |
Copyright terms: Public domain | W3C validator |