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Mirrors > Home > MPE Home > Th. List > hbab1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of hbab1 2718 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 2710 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
2 | hbs1 2265 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | hbxfrbi 1827 | 1 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 [wsb 2067 ∈ wcel 2106 {cab 2709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 |
This theorem is referenced by: (None) |
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