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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-rgenw | Structured version Visualization version GIF version |
Description: Generalization rule for restricted quantification. (Contributed by Wolf Lammen, 10-Jun-2023.) |
Ref | Expression |
---|---|
wl-rgenw.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
wl-rgenw | ⊢ ∀(𝑥 : 𝐴)𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-dfralsb 35002 | . 2 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑)) | |
2 | wl-rgenw.1 | . . . 4 ⊢ 𝜑 | |
3 | 2 | sbt 2071 | . . 3 ⊢ [𝑧 / 𝑥]𝜑 |
4 | 3 | a1i 11 | . 2 ⊢ (𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑) |
5 | 1, 4 | mpgbir 1801 | 1 ⊢ ∀(𝑥 : 𝐴)𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2069 ∈ wcel 2111 ∀wl-ral 34996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-wl-ral 35001 |
This theorem is referenced by: wl-rgen2w 35009 |
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