Theorem r19.29af 3245

Description: A commonly used pattern based on r19.29 3099. See r19.29a 3144, r19.29an 3140 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 3244	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1784   ∈ wcel 2113  ∃wrex 3060

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 2184

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-ral 3052  df-rex 3061

This theorem is referenced by:  fsnex  7229  neiptopnei  23076  neitr  23124  utopsnneiplem  24191  isucn2  24222  2sqmo  27404  foresf1o  32579  fsumiunle  32910  nsgqusf1olem3  33496  irngnzply1  33848  reff  33996  locfinreflem  33997  ordtconnlem1  34081  esumrnmpt2  34225  esumgect  34247  esum2dlem  34249  esum2d  34250  esumiun  34251  sigapildsys  34319  oms0  34454  eulerpartlemgvv  34533  breprexplema  34787  stoweidlem27  46271  stoweidlem35  46279