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Theorem pm10.541 37387
Description: Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm10.541 (∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑𝜓)))
Distinct variable group:   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem pm10.541
StepHypRef Expression
1 bi2.04 374 . . . 4 ((𝜑 → (¬ 𝜒𝜓)) ↔ (¬ 𝜒 → (𝜑𝜓)))
21albii 1735 . . 3 (∀𝑥(𝜑 → (¬ 𝜒𝜓)) ↔ ∀𝑥𝜒 → (𝜑𝜓)))
3 19.21v 1853 . . 3 (∀𝑥𝜒 → (𝜑𝜓)) ↔ (¬ 𝜒 → ∀𝑥(𝜑𝜓)))
42, 3bitri 262 . 2 (∀𝑥(𝜑 → (¬ 𝜒𝜓)) ↔ (¬ 𝜒 → ∀𝑥(𝜑𝜓)))
5 df-or 383 . . . 4 ((𝜒𝜓) ↔ (¬ 𝜒𝜓))
65imbi2i 324 . . 3 ((𝜑 → (𝜒𝜓)) ↔ (𝜑 → (¬ 𝜒𝜓)))
76albii 1735 . 2 (∀𝑥(𝜑 → (𝜒𝜓)) ↔ ∀𝑥(𝜑 → (¬ 𝜒𝜓)))
8 df-or 383 . 2 ((𝜒 ∨ ∀𝑥(𝜑𝜓)) ↔ (¬ 𝜒 → ∀𝑥(𝜑𝜓)))
94, 7, 83bitr4i 290 1 (∀𝑥(𝜑 → (𝜒𝜓)) ↔ (𝜒 ∨ ∀𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825
This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695
This theorem is referenced by: (None)
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