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| Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) | 
| Ref | Expression | 
|---|---|
| hb.1 | ⊢ (𝜑 → ∀𝑥𝜑) | 
| hb.2 | ⊢ (𝜓 → ∀𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| hban | ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hb.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
| 2 | 1 | nf5i 2146 | . . 3 ⊢ Ⅎ𝑥𝜑 | 
| 3 | hb.2 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
| 4 | 3 | nf5i 2146 | . . 3 ⊢ Ⅎ𝑥𝜓 | 
| 5 | 2, 4 | nfan 1899 | . 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) | 
| 6 | 5 | nf5ri 2195 | 1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: bnj982 34792 bnj1351 34840 bnj1352 34841 bnj1441 34854 bnj1441g 34855 copsex2b 37141 dvelimf-o 38930 ax12indalem 38946 ax12inda2ALT 38947 hbimpg 44574 hbimpgVD 44924 | 
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