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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 3127 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → (𝑥 ∈ 𝐴 → 𝜓)) | |
3 | 1, 2 | sylbi 220 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) |
4 | 3 | impcom 411 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2110 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-ral 3066 |
This theorem is referenced by: bnj229 32577 bnj999 32651 |
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