Theorem r19.29af 3267

Description: A commonly used pattern based on r19.29 3113. See r19.29a 3161, r19.29an 3157 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 1913	. 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 3266	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1782   ∈ wcel 2107  ∃wrex 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-ral 3061  df-rex 3070

This theorem is referenced by:  fsnex  7304  neiptopnei  23141  neitr  23189  utopsnneiplem  24257  isucn2  24289  2sqmo  27482  foresf1o  32524  fsumiunle  32832  nsgqusf1olem3  33444  irngnzply1  33742  reff  33839  locfinreflem  33840  ordtconnlem1  33924  esumrnmpt2  34070  esumgect  34092  esum2dlem  34094  esum2d  34095  esumiun  34096  sigapildsys  34164  oms0  34300  eulerpartlemgvv  34379  breprexplema  34646  stoweidlem27  46047  stoweidlem35  46055