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Mirrors > Home > MPE Home > Th. List > hbabg | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2390. See hbab 2810 for a version with more disjoint variable conditions, but not requiring ax-13 2390. (Contributed by NM, 1-Mar-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hbabg.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbabg | ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2800 | . 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
2 | hbabg.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 2 | hbsb 2567 | . 2 ⊢ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑) |
4 | 1, 3 | hbxfrbi 1825 | 1 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 [wsb 2069 ∈ wcel 2114 {cab 2799 |
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 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 |
This theorem is referenced by: nfsabg 2813 bnj1441g 32113 |
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