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Theorem hblem 2717
Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2743. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypothesis
Ref Expression
hblem.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Assertion
Ref Expression
hblem (𝑧𝐴 → ∀𝑥 𝑧𝐴)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem hblem
StepHypRef Expression
1 hblem.1 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
21hbsb 2428 . 2 ([𝑧 / 𝑦]𝑦𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦𝐴)
3 clelsb3 2715 . 2 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43albii 1736 . 2 (∀𝑥[𝑧 / 𝑦]𝑦𝐴 ↔ ∀𝑥 𝑧𝐴)
52, 3, 43imtr3i 278 1 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  [wsb 1866  wcel 1976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-cleq 2602  df-clel 2605
This theorem is referenced by:  nfcrii  2743  bnj1311  30152
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