Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-rgen | Structured version Visualization version GIF version |
Description: Generalization rule for restricted quantification. (Contributed by Wolf Lammen, 10-Jun-2023.) |
Ref | Expression |
---|---|
wl-rgen.1 | ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Ref | Expression |
---|---|
wl-rgen | ⊢ ∀(𝑥 : 𝐴)𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-dfralv 34877 | . 2 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
2 | wl-rgen.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝜑) | |
3 | 1, 2 | mpgbir 1799 | 1 ⊢ ∀(𝑥 : 𝐴)𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ∀wl-ral 34867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-11 2160 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-clel 2892 df-wl-ral 34872 |
This theorem is referenced by: wl-ralel 34881 |
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