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Mirrors > Home > MPE Home > Th. List > hbral | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.) |
Ref | Expression |
---|---|
hbral.1 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
hbral.2 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbral | ⊢ (∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥∀𝑦 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3061 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) | |
2 | hbral.1 | . . . 4 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
3 | hbral.2 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
4 | 2, 3 | hbim 2294 | . . 3 ⊢ ((𝑦 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑦 ∈ 𝐴 → 𝜑)) |
5 | 4 | hbal 2166 | . 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝜑) → ∀𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) |
6 | 1, 5 | hbxfrbi 1826 | 1 ⊢ (∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥∀𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1538 ∈ wcel 2105 ∀wral 3060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-11 2153 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-ex 1781 df-nf 1785 df-ral 3061 |
This theorem is referenced by: nfralw 3307 tratrbVD 44085 |
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