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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 2389. See hblem 2942 for a version with more disjoint variable conditions, but not requiring ax-13 2389. (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 2566 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴) |
3 | clelsb3 2939 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) | |
4 | 3 | albii 1819 | . 2 ⊢ (∀𝑥[𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑧 ∈ 𝐴) |
5 | 2, 3, 4 | 3imtr3i 293 | 1 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 [wsb 2068 ∈ wcel 2113 |
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 1969 ax-7 2014 ax-8 2115 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clel 2892 |
This theorem is referenced by: (None) |
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