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Mirrors > Home > MPE Home > Th. List > hblemg | Structured version Visualization version GIF version |
Description: Change the free variable of a hypothesis builder. Usage of this theorem is discouraged because it depends on ax-13 2370. See hblem 2864 for a version with more disjoint variable conditions, but not requiring ax-13 2370. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hblemg.1 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Ref | Expression |
---|---|
hblemg | ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hblemg.1 | . . 3 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
2 | 1 | hbsb 2522 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴) |
3 | clelsb1 2859 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) | |
4 | 3 | albii 1821 | . 2 ⊢ (∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑧 ∈ 𝐴) |
5 | 2, 3, 4 | 3imtr3i 290 | 1 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 [wsb 2067 ∈ wcel 2106 |
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-8 2108 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clel 2809 |
This theorem is referenced by: (None) |
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