MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.29af Structured version   Visualization version   GIF version

Theorem r19.29af 3214
Description: A commonly used pattern based on r19.29 3210. (Contributed by Thierry Arnoux, 29-Nov-2017.)
Hypotheses
Ref Expression
r19.29af.0 𝑥𝜑
r19.29af.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
r19.29af.2 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
r19.29af (𝜑𝜒)
Distinct variable group:   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.29af
StepHypRef Expression
1 r19.29af.0 . 2 𝑥𝜑
2 nfv 1992 . 2 𝑥𝜒
3 r19.29af.1 . 2 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
4 r19.29af.2 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
51, 2, 3, 4r19.29af2 3213 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wnf 1857  wcel 2139  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-ral 3055  df-rex 3056
This theorem is referenced by:  r19.29an  3215  r19.29a  3216  elsnxpOLD  5839  fsnex  6701  neiptopnei  21138  neitr  21186  utopsnneiplem  22252  isucn2  22284  foresf1o  29650  fsumiunle  29884  2sqmo  29958  reff  30215  locfinreflem  30216  ordtconnlem1  30279  esumrnmpt2  30439  esumgect  30461  esum2dlem  30463  esum2d  30464  esumiun  30465  sigapildsys  30534  oms0  30668  eulerpartlemgvv  30747  breprexplema  31017  stoweidlem27  40747  stoweidlem35  40755
  Copyright terms: Public domain W3C validator