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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.) |
Ref | Expression |
---|---|
hbab.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbab | ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2597 | . 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
2 | hbab.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 2 | hbsb 2429 | . 2 ⊢ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑) |
4 | 1, 3 | hbxfrbi 1742 | 1 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 [wsb 1867 ∈ wcel 1977 {cab 2596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 |
This theorem is referenced by: nfsab 2602 bnj1441 30165 bnj1309 30344 |
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