Theorem r19.29af 3266

Description: A commonly used pattern based on r19.29 3112. See r19.29a 3160, r19.29an 3156 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 1912	. 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 395  Ⅎwnf 1780   ∈ wcel 2106  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-ral 3060  df-rex 3069

This theorem is referenced by:  fsnex  7303  neiptopnei  23156  neitr  23204  utopsnneiplem  24272  isucn2  24304  2sqmo  27496  foresf1o  32532  fsumiunle  32836  nsgqusf1olem3  33423  irngnzply1  33706  reff  33800  locfinreflem  33801  ordtconnlem1  33885  esumrnmpt2  34049  esumgect  34071  esum2dlem  34073  esum2d  34074  esumiun  34075  sigapildsys  34143  oms0  34279  eulerpartlemgvv  34358  breprexplema  34624  stoweidlem27  45983  stoweidlem35  45991