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Mirrors > Home > ILE Home > Th. List > hb3or | 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.) |
Ref | Expression |
---|---|
hb.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hb.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
hb.3 | ⊢ (𝜒 → ∀𝑥𝜒) |
Ref | Expression |
---|---|
hb3or | ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → ∀𝑥(𝜑 ∨ 𝜓 ∨ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3or 920 | . 2 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒)) | |
2 | hb.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | hb.2 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 2, 3 | hbor 1478 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
5 | hb.3 | . . 3 ⊢ (𝜒 → ∀𝑥𝜒) | |
6 | 4, 5 | hbor 1478 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) → ∀𝑥((𝜑 ∨ 𝜓) ∨ 𝜒)) |
7 | 1, 6 | hbxfrbi 1401 | 1 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → ∀𝑥(𝜑 ∨ 𝜓 ∨ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 661 ∨ w3o 918 ∀wal 1282 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-gen 1378 |
This theorem depends on definitions: df-bi 115 df-3or 920 |
This theorem is referenced by: (None) |
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