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Mirrors > Home > MPE Home > Th. List > nfan | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ∧ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 9-Oct-2021.) |
Ref | Expression |
---|---|
nfan.1 | ⊢ Ⅎ𝑥𝜑 |
nfan.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
nfan | ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfan.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
3 | nfan.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜓) |
5 | 2, 4 | nfand 1905 | . 2 ⊢ (⊤ → Ⅎ𝑥(𝜑 ∧ 𝜓)) |
6 | 5 | mptru 1550 | 1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) |
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