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| 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 2150 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
| 2 | 1 | nf5ri 2194 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-or 848 df-ex 1779 df-nf 1783 | 
| This theorem is referenced by: axi5r 2699 axial 2700 bj-19.41al 36661 bj-wnf1 36719 hbntal 44578 hbimpg 44579 hbimpgVD 44929 hbalgVD 44930 hbexgVD 44931 ax6e2eqVD 44932 e2ebindVD 44937 vk15.4jVD 44939 | 
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