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Theorem hban 1509
Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
Hypotheses
Ref Expression
hb.1 (𝜑 → ∀𝑥𝜑)
hb.2 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
hban ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Proof of Theorem hban
StepHypRef Expression
1 hb.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 hb.2 . . 3 (𝜓 → ∀𝑥𝜓)
31, 2anim12i 334 . 2 ((𝜑𝜓) → (∀𝑥𝜑 ∧ ∀𝑥𝜓))
4 19.26 1440 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
53, 4sylibr 133 1 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  hbbi  1510  hb3an  1512  hbsbv  1892  mopick  2053  eupicka  2055  mopick2  2058  cleqh  2215
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