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Theorem bj-hblem 32043
 Description: Remove dependency on ax-ext 2590 (and df-cleq 2603) from hblem 2718. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-hblem.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Assertion
Ref Expression
bj-hblem (𝑧𝐴 → ∀𝑥 𝑧𝐴)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem bj-hblem
StepHypRef Expression
1 bj-hblem.1 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
21hbsb 2429 . 2 ([𝑧 / 𝑦]𝑦𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦𝐴)
3 bj-clelsb3 32042 . 2 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43albii 1737 . 2 (∀𝑥[𝑧 / 𝑦]𝑦𝐴 ↔ ∀𝑥 𝑧𝐴)
52, 3, 43imtr3i 279 1 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  [wsb 1867   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clel 2606 This theorem is referenced by:  bj-nfcrii  32045
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