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Theorem 19.21vv 37395
Description: Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1853. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
19.21vv (∀𝑥𝑦(𝜓𝜑) ↔ (𝜓 → ∀𝑥𝑦𝜑))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.21vv
StepHypRef Expression
1 19.21v 1853 . . 3 (∀𝑦(𝜓𝜑) ↔ (𝜓 → ∀𝑦𝜑))
21albii 1735 . 2 (∀𝑥𝑦(𝜓𝜑) ↔ ∀𝑥(𝜓 → ∀𝑦𝜑))
3 19.21v 1853 . 2 (∀𝑥(𝜓 → ∀𝑦𝜑) ↔ (𝜓 → ∀𝑥𝑦𝜑))
42, 3bitri 262 1 (∀𝑥𝑦(𝜓𝜑) ↔ (𝜓 → ∀𝑥𝑦𝜑))
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 1711  ax-4 1726  ax-5 1825
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by: (None)
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