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Mirrors > Home > MPE Home > Th. List > nf3and | Structured version Visualization version GIF version |
Description: Deduction form of bound-variable hypothesis builder nf3an 1897. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.) |
Ref | Expression |
---|---|
nfand.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
nfand.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
nfand.3 | ⊢ (𝜑 → Ⅎ𝑥𝜃) |
Ref | Expression |
---|---|
nf3and | ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒 ∧ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1087 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃)) | |
2 | nfand.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | nfand.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | 2, 3 | nfand 1893 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒)) |
5 | nfand.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜃) | |
6 | 4, 5 | nfand 1893 | . 2 ⊢ (𝜑 → Ⅎ𝑥((𝜓 ∧ 𝜒) ∧ 𝜃)) |
7 | 1, 6 | nfxfrd 1849 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒 ∧ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-ex 1775 df-nf 1779 |
This theorem is referenced by: nfttrcld 9734 |
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