Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2stdpc5 Structured version   Visualization version   GIF version

Theorem 2stdpc5 34283
 Description: A double stdpc5 2206 (one direction of PM*11.3). See also 2stdpc4 2075 and 19.21vv 41095. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
2stdpc5.1 𝑥𝜑
2stdpc5.2 𝑦𝜑
Assertion
Ref Expression
2stdpc5 (∀𝑥𝑦(𝜑𝜓) → (𝜑 → ∀𝑥𝑦𝜓))

Proof of Theorem 2stdpc5
StepHypRef Expression
1 2stdpc5.2 . . . 4 𝑦𝜑
21stdpc5 2206 . . 3 (∀𝑦(𝜑𝜓) → (𝜑 → ∀𝑦𝜓))
32alimi 1813 . 2 (∀𝑥𝑦(𝜑𝜓) → ∀𝑥(𝜑 → ∀𝑦𝜓))
4 2stdpc5.1 . . 3 𝑥𝜑
54stdpc5 2206 . 2 (∀𝑥(𝜑 → ∀𝑦𝜓) → (𝜑 → ∀𝑥𝑦𝜓))
63, 5syl 17 1 (∀𝑥𝑦(𝜑𝜓) → (𝜑 → ∀𝑥𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786 This theorem is referenced by:  ax11-pm  34286  ax11-pm2  34290
 Copyright terms: Public domain W3C validator