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Mirrors > Home > ILE Home > Th. List > r19.32vr | GIF version |
Description: One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence, as seen at r19.32vdc 2583. (Contributed by Jim Kingdon, 19-Aug-2018.) |
Ref | Expression |
---|---|
r19.32vr | ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1509 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.32r 2581 | 1 ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 ∀wral 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-gen 1426 ax-4 1488 ax-17 1507 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-ral 2422 |
This theorem is referenced by: iinuniss 3903 |
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