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Mirrors > Home > ILE Home > Th. List > hband | GIF version |
Description: Deduction form of bound-variable hypothesis builder hban 1439. (Contributed by NM, 2-Jan-2002.) |
Ref | Expression |
---|---|
hband.1 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
hband.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
Ref | Expression |
---|---|
hband | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hband.1 | . . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
2 | hband.2 | . . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
3 | 1, 2 | anim12d 318 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜒))) |
4 | 19.26 1370 | . 2 ⊢ (∀𝑥(𝜓 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜒)) | |
5 | 3, 4 | syl6ibr 151 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀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 |
This theorem is referenced by: (None) |
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