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Mirrors > Home > ILE Home > Th. List > hban | GIF version |
Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.) |
Ref | Expression |
---|---|
hb.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hb.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
hban | ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hb.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | hb.2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | 1, 2 | anim12i 331 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (∀𝑥𝜑 ∧ ∀𝑥𝜓)) |
4 | 19.26 1410 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | |
5 | 3, 4 | sylibr 132 | 1 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀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-5 1376 ax-gen 1378 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: hbbi 1480 hb3an 1482 hbsbv 1858 mopick 2019 eupicka 2021 mopick2 2024 cleqh 2178 |
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