Theorem r19.29af 3246

Description: A commonly used pattern based on r19.29 3100. See r19.29a 3145, r19.29an 3141 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 1916	. 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 3245	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1785   ∈ wcel 2114  ∃wrex 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-ral 3052  df-rex 3062

This theorem is referenced by:  fsnex  7238  neiptopnei  23097  neitr  23145  utopsnneiplem  24212  isucn2  24243  2sqmo  27400  foresf1o  32574  fsumiunle  32902  nsgqusf1olem3  33475  irngnzply1  33835  reff  33983  locfinreflem  33984  ordtconnlem1  34068  esumrnmpt2  34212  esumgect  34234  esum2dlem  34236  esum2d  34237  esumiun  34238  sigapildsys  34306  oms0  34441  eulerpartlemgvv  34520  breprexplema  34774  stoweidlem27  46455  stoweidlem35  46463