Theorem r19.29af 3247

Description: A commonly used pattern based on r19.29 3095. See r19.29a 3142, r19.29an 3138 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 1914	. 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 3246	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1783   ∈ wcel 2109  ∃wrex 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-ral 3046  df-rex 3055

This theorem is referenced by:  fsnex  7261  neiptopnei  23026  neitr  23074  utopsnneiplem  24142  isucn2  24173  2sqmo  27355  foresf1o  32440  fsumiunle  32761  nsgqusf1olem3  33393  irngnzply1  33693  reff  33836  locfinreflem  33837  ordtconnlem1  33921  esumrnmpt2  34065  esumgect  34087  esum2dlem  34089  esum2d  34090  esumiun  34091  sigapildsys  34159  oms0  34295  eulerpartlemgvv  34374  breprexplema  34628  stoweidlem27  46032  stoweidlem35  46040