Theorem r19.29af 3280

Description: A commonly used pattern based on r19.29 3134. See r19.29a 3179, r19.29an 3175 for a variant when 𝑥 is disjoint from 𝜑. (Contributed by Thierry Arnoux, 29-Nov-2017.)

Hypotheses

Ref	Expression
r19.29af.0	⊢ Ⅎ𝑥𝜑
r19.29af.1	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
r19.29af.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
r19.29af	⊢ (𝜑 → 𝜒)

Distinct variable group:   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.29af

Step	Hyp	Ref	Expression
1		r19.29af.0	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1941	. 2 ⊢ Ⅎ𝑥𝜒
3		r19.29af.1	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
4		r19.29af.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
5	1, 2, 3, 4	r19.29af2 3279	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 400  Ⅎwnf 1810   ∈ wcel 2149  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811  df-ral 3086  df-rex 3096

This theorem is referenced by:  fsnex  7282  neiptopnei  23257  neitr  23305  utopsnneiplem  24372  isucn2  24403  2sqmo  27566  foresf1o  32790  fsumiunle  33113  nsgqusf1olem3  33667  irngnzply1  34025  reff  34173  locfinreflem  34174  ordtconnlem1  34258  esumrnmpt2  34402  esumgect  34424  esum2dlem  34426  esum2d  34427  esumiun  34428  sigapildsys  34496  oms0  34631  eulerpartlemgvv  34710  breprexplema  34961  stoweidlem27  46632  stoweidlem35  46640