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Mirrors > Home > MPE Home > Th. List > hbab | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2375. See hbabg 2724 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) |
Ref | Expression |
---|---|
hbab.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbab | ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2713 | . 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
2 | hbab.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 2 | hbsbw 2169 | . 2 ⊢ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑) |
4 | 1, 3 | hbxfrbi 1822 | 1 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 [wsb 2062 ∈ wcel 2106 {cab 2712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-11 2155 |
This theorem depends on definitions: df-bi 207 df-ex 1777 df-sb 2063 df-clab 2713 |
This theorem is referenced by: nfsab 2725 bnj1441 34833 bnj1309 35015 |
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