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| Mirrors > Home > MPE Home > Th. List > hbab1OLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of hbab1 2723 as of 25-Oct-2024. (Contributed by NM, 26-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hbab1OLD | ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-clab 2715 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
| 2 | hbs1 2274 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
| 3 | 1, 2 | hbxfrbi 1825 | 1 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 [wsb 2064 ∈ wcel 2108 {cab 2714 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 | 
| This theorem is referenced by: (None) | 
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