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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1459 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1459.1 | ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴)) |
bnj1459.2 | ⊢ (𝜓 → 𝜒) |
Ref | Expression |
---|---|
bnj1459 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1459.1 | . . 3 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴)) | |
2 | bnj1459.2 | . . 3 ⊢ (𝜓 → 𝜒) | |
3 | 1, 2 | sylbir 234 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜒) |
4 | 3 | ralrimiva 3108 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ral 3069 |
This theorem is referenced by: bnj1501 33055 |
Copyright terms: Public domain | W3C validator |