MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.31 Structured version   Visualization version   GIF version

Theorem 19.31 2087
Description: Theorem 19.31 of [Margaris] p. 90. See 19.31v 1856 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.)
Hypothesis
Ref Expression
19.31.1 𝑥𝜓
Assertion
Ref Expression
19.31 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Proof of Theorem 19.31
StepHypRef Expression
1 19.31.1 . . 3 𝑥𝜓
2119.32 2086 . 2 (∀𝑥(𝜓𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑))
3 orcom 400 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
43albii 1736 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥(𝜓𝜑))
5 orcom 400 . 2 ((∀𝑥𝜑𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑))
62, 4, 53bitr4i 290 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wo 381  wal 1472  wnf 1698
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  ax-6 1874  ax-7 1921  ax-12 2032
This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695  df-nf 1700
This theorem is referenced by:  2eu3  2542
  Copyright terms: Public domain W3C validator