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Mirrors > Home > ILE Home > Th. List > nf3and | GIF version |
Description: Deduction form of bound-variable hypothesis builder nf3an 1566. (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 980 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃)) | |
2 | nfand.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | nfand.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | 2, 3 | nfand 1568 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒)) |
5 | nfand.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜃) | |
6 | 4, 5 | nfand 1568 | . 2 ⊢ (𝜑 → Ⅎ𝑥((𝜓 ∧ 𝜒) ∧ 𝜃)) |
7 | 1, 6 | nfxfrd 1475 | 1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒 ∧ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 Ⅎwnf 1460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-nf 1461 |
This theorem is referenced by: (None) |
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