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Theorem hbim 2112
 Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
Hypotheses
Ref Expression
hbim.1 (𝜑 → ∀𝑥𝜑)
hbim.2 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
hbim ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Proof of Theorem hbim
StepHypRef Expression
1 hbim.1 . 2 (𝜑 → ∀𝑥𝜑)
2 hbim.2 . . 3 (𝜓 → ∀𝑥𝜓)
32a1i 11 . 2 (𝜑 → (𝜓 → ∀𝑥𝜓))
41, 3hbim1 2110 1 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473 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-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  axi5r  2582  hbral  2927
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