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Theorem r19.29an 3070
Description: A commonly used pattern based on r19.29 3065. (Contributed by Thierry Arnoux, 29-Dec-2019.)
Hypothesis
Ref Expression
r19.29an.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
Assertion
Ref Expression
r19.29an ((𝜑 ∧ ∃𝑥𝐴 𝜓) → 𝜒)
Distinct variable groups:   𝜒,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.29an
StepHypRef Expression
1 nfv 1840 . . 3 𝑥𝜑
2 nfre1 2999 . . 3 𝑥𝑥𝐴 𝜓
31, 2nfan 1825 . 2 𝑥(𝜑 ∧ ∃𝑥𝐴 𝜓)
4 r19.29an.1 . . 3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
54adantllr 754 . 2 ((((𝜑 ∧ ∃𝑥𝐴 𝜓) ∧ 𝑥𝐴) ∧ 𝜓) → 𝜒)
6 simpr 477 . 2 ((𝜑 ∧ ∃𝑥𝐴 𝜓) → ∃𝑥𝐴 𝜓)
73, 5, 6r19.29af 3069 1 ((𝜑 ∧ ∃𝑥𝐴 𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-ral 2912  df-rex 2913
This theorem is referenced by:  summolem2  14380  cygabl  18213  dissnlocfin  21242  utopsnneiplem  21961  restmetu  22285  elqaa  23981  colline  25444  f1otrg  25651  axcontlem2  25745  grpoidinvlem4  27210  2sqmo  29434  isarchi3  29526  fimaproj  29682  qtophaus  29685  locfinreflem  29689  cmpcref  29699  ordtconnlem1  29752  esumpcvgval  29921  esumcvg  29929  eulerpartlems  30203  eulerpartlemgvv  30219  isbnd3  33215  eldiophss  36818  eldioph4b  36855  pellfund14b  36943  opeoALTV  40894
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