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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, 12-Oct-2021.) |
Ref | Expression |
---|---|
hba1 | ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2149 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
2 | 1 | nf5ri 2189 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-or 847 df-ex 1783 df-nf 1787 |
This theorem is referenced by: axi5r 2696 axial 2697 bj-19.41al 35536 bj-wnf1 35595 hbntal 43314 hbimpg 43315 hbimpgVD 43665 hbalgVD 43666 hbexgVD 43667 ax6e2eqVD 43668 e2ebindVD 43673 vk15.4jVD 43675 |
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