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Mirrors > Home > ILE Home > Th. List > hb3and | GIF version |
Description: Deduction form of bound-variable hypothesis builder hb3an 1442. (Contributed by NM, 17-Feb-2013.) |
Ref | Expression |
---|---|
hb3and.1 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
hb3and.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
hb3and.3 | ⊢ (𝜑 → (𝜃 → ∀𝑥𝜃)) |
Ref | Expression |
---|---|
hb3and | ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜓 ∧ 𝜒 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hb3and.1 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
2 | hb3and.2 | . . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
3 | hb3and.3 | . . 3 ⊢ (𝜑 → (𝜃 → ∀𝑥𝜃)) | |
4 | 1, 2, 3 | 3anim123d 1214 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → (∀𝑥𝜓 ∧ ∀𝑥𝜒 ∧ ∀𝑥𝜃))) |
5 | 19.26-3an 1372 | . 2 ⊢ (∀𝑥(𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒 ∧ ∀𝑥𝜃)) | |
6 | 4, 5 | syl6ibr 151 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜓 ∧ 𝜒 ∧ 𝜃))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 885 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 |
This theorem depends on definitions: df-bi 110 df-3an 887 |
This theorem is referenced by: (None) |
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