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Theorem hbim 1817
 Description: If x is not free in φ and ψ, it is not free in (φ → ψ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
Hypotheses
Ref Expression
hbim.1 (φxφ)
hbim.2 (ψxψ)
Assertion
Ref Expression
hbim ((φψ) → x(φψ))

Proof of Theorem hbim
StepHypRef Expression
1 hbim.1 . 2 (φxφ)
2 hbim.2 . . 3 (ψxψ)
32a1i 10 . 2 (φ → (ψxψ))
41, 3hbim1 1810 1 ((φψ) → x(φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545 This theorem is referenced by:  19.23hOLD  1820  hbanOLD  1829  19.21hOLD  1840  cbv3hvOLD  1851  ax12olem5  1931  axi5r  2326  cleqh  2450  hbral  2662
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