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Mirrors > Home > MPE Home > Th. List > hbab1 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) |
Ref | Expression |
---|---|
hbab1 | ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2760 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
2 | hbs1 2203 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | hbxfrbi 1787 | 1 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1505 [wsb 2015 ∈ wcel 2050 {cab 2759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-10 2079 ax-12 2106 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 |
This theorem is referenced by: nfsab1OLD 2769 abeq2 2898 abbi 2908 abeq2fOLD 2966 bnj1317 31738 bnj1318 31939 |
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