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Theorem pm10.542 37392
Description: Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.542 (∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 → ∀𝑥(𝜑𝜓)))
Distinct variable group:   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem pm10.542
StepHypRef Expression
1 bi2.04 374 . . 3 ((𝜑 → (𝜒𝜓)) ↔ (𝜒 → (𝜑𝜓)))
21albii 1736 . 2 (∀𝑥(𝜑 → (𝜒𝜓)) ↔ ∀𝑥(𝜒 → (𝜑𝜓)))
3 19.21v 1854 . 2 (∀𝑥(𝜒 → (𝜑𝜓)) ↔ (𝜒 → ∀𝑥(𝜑𝜓)))
42, 3bitri 262 1 (∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 → ∀𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472
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
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by: (None)
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