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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj228 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj228.1 | ⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓) |
Ref | Expression |
---|---|
bnj228 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj228.1 | . . 3 ⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓) | |
2 | rsp 3202 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → (𝑥 ∈ 𝐴 → 𝜓)) | |
3 | 1, 2 | sylbi 218 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) |
4 | 3 | impcom 408 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-ral 3140 |
This theorem is referenced by: bnj229 32055 bnj999 32128 |
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