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Theorem hblemg 2882
 Description: Change the free variable of a hypothesis builder. Usage of this theorem is discouraged because it depends on ax-13 2380. See hblem 2881 for a version with more disjoint variable conditions, but not requiring ax-13 2380. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
hblemg.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Assertion
Ref Expression
hblemg (𝑧𝐴 → ∀𝑥 𝑧𝐴)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem hblemg
StepHypRef Expression
1 hblemg.1 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
21hbsb 2545 . 2 ([𝑧 / 𝑦]𝑦𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦𝐴)
3 clelsb3 2878 . 2 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43albii 1822 . 2 (∀𝑥[𝑧 / 𝑦]𝑦𝐴 ↔ ∀𝑥 𝑧𝐴)
52, 3, 43imtr3i 295 1 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1537  [wsb 2070   ∈ wcel 2112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clel 2831 This theorem is referenced by: (None)
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