Theorem r19.29af 3241

Description: A commonly used pattern based on r19.29 3095. See r19.29a 3140, r19.29an 3136 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 1915	. 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 3240	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2111  ∃wrex 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-ral 3048  df-rex 3057

This theorem is referenced by:  fsnex  7217  neiptopnei  23047  neitr  23095  utopsnneiplem  24162  isucn2  24193  2sqmo  27375  foresf1o  32484  fsumiunle  32812  nsgqusf1olem3  33380  irngnzply1  33704  reff  33852  locfinreflem  33853  ordtconnlem1  33937  esumrnmpt2  34081  esumgect  34103  esum2dlem  34105  esum2d  34106  esumiun  34107  sigapildsys  34175  oms0  34310  eulerpartlemgvv  34389  breprexplema  34643  stoweidlem27  46073  stoweidlem35  46081