Theorem r19.29af 3266

Description: A commonly used pattern based on r19.29 3115. See r19.29a 3163, r19.29an 3159 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 1918	. 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 3265	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 397  Ⅎwnf 1786   ∈ wcel 2107  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-nf 1787  df-ral 3063  df-rex 3072

This theorem is referenced by:  fsnex  7281  neiptopnei  22636  neitr  22684  utopsnneiplem  23752  isucn2  23784  2sqmo  26940  foresf1o  31742  fsumiunle  32035  nsgqusf1olem3  32526  irngnzply1  32755  reff  32819  locfinreflem  32820  ordtconnlem1  32904  esumrnmpt2  33066  esumgect  33088  esum2dlem  33090  esum2d  33091  esumiun  33092  sigapildsys  33160  oms0  33296  eulerpartlemgvv  33375  breprexplema  33642  stoweidlem27  44743  stoweidlem35  44751