MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.21 Structured version   Visualization version   GIF version

Theorem 19.21 2239
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 2034 for a version requiring fewer axioms. See also 19.21h 2297. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1879 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
Hypothesis
Ref Expression
19.21.1 𝑥𝜑
Assertion
Ref Expression
19.21 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Proof of Theorem 19.21
StepHypRef Expression
1 19.21.1 . 2 𝑥𝜑
2 19.21t 2238 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
31, 2ax-mp 5 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-ex 1875  df-nf 1879
This theorem is referenced by:  stdpc5  2240  19.21-2  2241  19.32  2266  nf6  2293  19.21h  2297  19.12vv  2344  cbv1  2372  axc14  2463  r2alf  3085  19.12b  32150  bj-biexal2  33132  bj-bialal  33134  bj-cbv1v  33163  wl-dral1d  33743  mpt2bi123f  34392
  Copyright terms: Public domain W3C validator