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Mirrors > Home > MPE Home > Th. List > hb3an | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
Ref | Expression |
---|---|
hb.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hb.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
hb.3 | ⊢ (𝜒 → ∀𝑥𝜒) |
Ref | Expression |
---|---|
hb3an | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hb.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | nf5i 2142 | . . 3 ⊢ Ⅎ𝑥𝜑 |
3 | hb.2 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 3 | nf5i 2142 | . . 3 ⊢ Ⅎ𝑥𝜓 |
5 | hb.3 | . . . 4 ⊢ (𝜒 → ∀𝑥𝜒) | |
6 | 5 | nf5i 2142 | . . 3 ⊢ Ⅎ𝑥𝜒 |
7 | 2, 4, 6 | nf3an 1904 | . 2 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) |
8 | 7 | nf5ri 2188 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∀wal 1537 |
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 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-ex 1783 df-nf 1787 |
This theorem is referenced by: bnj982 32766 bnj1276 32802 bnj1350 32813 |
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