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Theorem 19.26-2 1387
Description: Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
Assertion
Ref Expression
19.26-2 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓))

Proof of Theorem 19.26-2
StepHypRef Expression
1 19.26 1386 . . 3 (∀𝑦(𝜑𝜓) ↔ (∀𝑦𝜑 ∧ ∀𝑦𝜓))
21albii 1375 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓))
3 19.26 1386 . 2 (∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓))
42, 3bitri 177 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  opelopabt  4027  fun11  4994
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